Linear Dependence of Vectors

Linear Independence and Dependence

Definition: A set of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = \{ \mathbf{x_1}, \mathbf{x_2}, ..., \mathbf{x_n} \}} and the scalars Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1, k_2, ..., k_n} has a solution for the vector equation $k_{1}\mathbf {x_{1}} +k_{2}\mathbf {x_{2}} +...+k_{n}\mathbf {x_{n}} =0$, namely when $k_{1}=k_{2}=...=k_{n}=0$. If this is the only solution to the vector equation, then the set $V$ is said to be Linearly Independent. If there exists other solutions where not all $k_{i}=0$, then $V$ is said to be Linearly Dependent.

We will now look at some examples of vector sets which are either linearly independent or linearly dependent.

Note: Another way to write that a set of vectors $\{v_{1},v_{2},...,v_{n}\}$ is linearly independent is by saying that $\mathrm {span} (v_{1},v_{2},...,v_{n})=\bigoplus _{i=1}^{n}\mathbb {F} v_{i}$, which says that any vector that is a linear combination of the set of vectors $v_{1},v_{2},...,v_{n}$ is a unique linear combination.

Example 1

Determine whether or not the vector set $V=\{(1,2),(-5,-3)\}$ is linearly independent or linearly dependent.

We first must see if there exists only one set of scalars $c_{1}=c_{2}=0$ (the trivial solution) as a solution to the vector equation $c_{1}(1,2)+c_{2}(-5,-3)=0$ or if there exists more solutions.

From this vector equation we get the following system of linear equations:

${\begin{aligned}c_{1}-5c_{2}=0\\2c_{1}-3c_{2}=0\end{aligned}}$

When we reduce this system to RREF, we get that:

${\begin{aligned}c_{1}+0c_{2}=0\\0c_{1}+c_{2}=0\end{aligned}}$

Therefore our only set of scalars $c_{1},c_{2}$ are both equal to zero. Therefore, the vector set $V$ is linearly independent.

Example 2

Determine whether the vector set $V=\{(3,6),(4,8)\}$ is a linearly independent or a linearly dependent set.

Once again, for $V$ to be a linearly independent set then the vector equation $c_{1}(3,6)+c_{2}(4,8)=0$ must containing only one set of scalars $c_{1}=c_{2}=0$. From our vector equation be obtain the following system of linear equations:

${\begin{aligned}3c_{1}+4c_{2}=0\\6c_{1}+8c_{2}=0\end{aligned}}$

When we solve this system of equations we obtain that:

${\begin{aligned}c_{1}+{\frac {4}{3}}c_{2}=0\\0c_{1}+0c_{2}=0\end{aligned}}$

Suppose that $c_{2}=s$ so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1 = - \frac{4}{3} s} . We see that for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s \in \mathbb{R}} , the scalars Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2} satisfy the vector equation and thus there are infinitely many scalars sets. Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is a linearly dependent set.

Example 3

Show that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} invertible matrix, then the column vectors of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_2} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_n} are linearly independent.

Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is an invertible matrix and suppose that we have the linear system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Ax = 0} . We note that the only solution to this linear system is the trivial solution, namely Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 0} . This system corresponds to the following vector equations though:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1C_1 + x_2C_2 + ... + x_nC_n = 0 }

And since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = (x_1, x_2, ..., x_n) = (0, 0, ..., 0)} , we see that this vector equation only have the scalars Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1 = x_2 = ... = x_n = 0} , so the column vectors of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} are linearly independent.

Example 4

Consider the set of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1, 1-x, 1-x^2 \}} from the vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \wp_2 (\mathbb{F})} . Determine if this set of vectors is linearly independent or linearly dependent.

When we expand the vector equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(1) + b(1 -x) + c(1 - x^2) = 0} as follows, notice:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} a(1) + b(1 -x) + c(1 - x^2) = 0 \\ a + b - bx + c - cx^2 = 0 \\ (a + b + c) + (-b)x + (-c)x^2 = 0 \end{align}}

We can clearly see that this equation holds only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = b = c = 0} . We can also verify this with the following matrix:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} A = \begin{bmatrix} 1 & 1 & 1\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \end{align}}

Since this is an upper triangular matrix, its determinant is the product of the entries down the main diagonal and so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{det} (A) = 1 \cdot (-1) \cdot (-1) = 1} which implies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is invertible, and so the homogenous system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix}1 & 1 & 1\\ 0 & -1 & 0\\ 0 & 0 & -1\end{bmatrix} \begin{bmatrix}a\\ b\\ c \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0 \end{bmatrix}} has only the trivial solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = b = c = 0} .

Example 5

Show that the set of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ m, m - x, m - x^2 \}} from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \wp_2 (\mathbb{F})} is a linearly independent set for all nonzero $m$.

We should note that this example is a more general version of example 4. First, let's expand the vector equation $a_{1}m+a_{2}(m-x)+a_{3}(m-x^{2})=0$ as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} a_1m + a_2(m - x) + a_3(m - x^2) = 0 \\ a_1m + a_2m - a_2x + a_3m - a_3x^2 = 0 \\ (a_1 + a_2 + a_3)m + (-a_2)x + (-a_3)x^2 = 0 \end{align}}

Once again, this vector equation is only true if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1 = a_2 = a_3 = 0} . The same matrix from example 4 can be used to provide a more extensive argument.

Linear Dependence Lemma

We will now look at a very important lemma known as the linear dependence lemma.

Lemma (Linear Dependence Lemma): Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ v_1, v_2, ..., v_m \}} be a set of linearly dependent vectors in the vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1 \neq 0} . Then there exists Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j \in \{ 2, 3, ..., m \}} such that:

a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_j \in \mathrm{span} \{ v_1, ..., v_{j-1} \}} .

b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{span} \{ v_1, ..., v_{j-1}, v_{j+1}, ..., v_m \} = \mathrm{span} \{ v_1, ..., v_m \}} .

The linear dependence lemma tells us that given a linear dependent set of vectors where the first vector is nonzero, then there exists a vector in the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ v_1, v_2, ..., v_m \}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_j} can be written as a linear combination of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ v_1, v_2, ..., v_{j-1}} (that is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_j \in \mathrm{span} \{ v_1, v_2, ..., v_{j-1} \}} ), and that the set of linear combinations of the set of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ v_1, ..., v_{j-1}, v_{j+1}, ..., v_m \}} is the same set as the combination of the set of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ v_1, v_2, ..., v_m \}} .

• Proof: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ v_1, v_2, ..., v_m \}} be a set of vectors that are linearly dependent where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1 \neq 0} , and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1, a_2, ..., a_m \in \mathbb{F}} . Since this set of vectors is not linearly independent, then:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1v_1 + a_2v_2 + ... + a_mv_m = 0 }

• Where not all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_i} are zero (otherwise the set of vectors would be linearly independent). Now since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1 \neq 0} , it follows that not all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_2, a_3, ..., a_m \in \mathbb{F}} are equal to zero, and so there exists an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_j \in \mathbb{F}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_j \neq 0} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} is the largest index such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_j \neq 0} , in other words, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{j+1} = 0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{j+2} = 0} , etc…, and so so:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad a_jv_j = -a_1v_1 - a_2v_2 - ... - a_{j-1}v_{j-1} - \underbrace{a_{j+1}v_{j+1}}_{=0} + ... + \underbrace{a_{m}v_{m}}_{=0} \\ \quad a_jv_j = -a_1v_1 - a_2v_2 - ... - a_{j-1}v_{j-1} \\ \quad v_j = -\frac{a_1}{a_j}v_1 -\frac{a_2}{a_j}v_2 - ... - \frac{a_{j-1}}{a_j}v_{j-1} \end{align}}

• Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_j} can be written as a linear combination of the set of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ v_1, v_2, ..., v_{j-1} \}} so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_j \in \mathrm{span} \{ v_1, v_2, ..., v_{j-1} \}} which proves A.

• Now let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u \in \mathrm{span} \{ v_1, v_2, ..., v_m \}} , and so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} can be written as a linear combination of the vectors in this set, and so there exists Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_1, b_2, ..., b_m \in \mathbb{F}} such that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u = b_1v_1 + b_2v_2 + ... + b_mv_m }

• Now we substitute that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \quad v_j = -\frac{a_1}{a_j}v_1 -\frac{a_2}{a_j}v_2 - ... - \frac{a_{j-1}}{a_j}v_{j-1}} and so:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad u = b_1v_1 + b_2v_2 + ... + b_{j-1}v_{j-1} + b_jv_j + b_{j+1}v_{j+1} + ... + b_mv_m \\ \quad u = b_1v_1 + b_2v_2 + ... + b_{j-1}v_{j-1} + b_j[-\frac{a_1}{a_j}v_1 -\frac{a_2}{a_j}v_2 - ... - \frac{a_{j-1}}{a_j}v_{j-1}] + b_{j+1}v_{j+1} + ... + b_mv_m \\ \quad u = (b_1 - a_1b_j)v_1 + (b_2 - a_2b_j)v_2 + ... + (b_{j-1} - a_{j-1}b_j)v_{j-1} + (b_{j+1} - a_{j+1}b_j)v_{j+1} + ... + (b_m - a_mb_j)v_m \end{align}}

• Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} can be written as a linear combination of the vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ v_1, ..., v_{j-1}, v_{j+1}, ..., v_m} and so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u \in \mathrm{span} \{ v_1, ..., v_{j-1}, v_{j+1}, ..., v_m \}} . Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{span} \{ v_1, v_2, ..., v_m \} = \mathrm{span} \{ v_1, ..., v_{j-1}, v_{j+1}, ..., v_m \}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}

Licensing

Content obtained and/or adapted from:

• Linear Independence and Dependence, mathonline.wikidot.com under a CC BY-SA license
• Linear Dependence Lemma, mathonline.wikidot.com under a CC BY-SA license