Vector and Matrix Equations

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Section 2.2 Vector and Matrix Equations

Recall that two vectors in \(\mathbb{R}^n\) are equal if and only if all of their components are equal.

Question 2.13.

Prove that the system of equations given by

\begin{equation*} \begin{array}{rcrcrcrcr} a_{11} x_1 \amp +\amp a_{12} x_2 \amp +\amp ... \amp +\amp a_{1n}x_n \amp =\amp b_1 \\ % -1 3 a_{21} x_1 \amp +\amp a_{22} x_2 \amp +\amp ... \amp +\amp a_{2n}x_n \amp =\amp b_2 \\ \vdots \amp \amp \vdots \amp \amp \amp \amp \vdots \amp \amp \vdots \\ a_{m1} x_1 \amp +\amp a_{m2} x_2 \amp +\amp ... \amp +\amp a_{mn}x_n \amp =\amp b_m \end{array} \end{equation*}

has the same set of solutions as the vector equation

\begin{equation*} x_1 \begin{bmatrix}a_{11}\\a_{21}\\\vdots\\a_{m1}\end{bmatrix}+x_2 \begin{bmatrix}a_{12}\\a_{22}\\\vdots\\a_{m2}\end{bmatrix}+ \cdots x_n \begin{bmatrix}a_{1n}\\a_{2n}\\\vdots\\a_{mn}\end{bmatrix}=\begin{bmatrix}b_1\\b_2\\\vdots\\b_m\end{bmatrix}\text{.} \end{equation*}

In other words, prove that \((c_1,c_2,...,c_n)\) is a solution to the vector equation iff \((c_1,c_2,...,c_n)\) is a solution to the system of linear equations. Make sure you clearly connect the ideas in your proof and do not make an argument that these equations look similar.

Question 2.14.

Solve the following vector equation directly:

\begin{equation*} x_1 \colvec{1\\1}+x_2 \colvec{-1\\1} = \colvec{3\\-4} \end{equation*}

Question 2.15.

Give an example of a vector \(\colvec{a\\b\\c}\) such that the equation

\begin{equation*} x_1\colvec{1\\1\\0}+x_2 \colvec{0\\0\\1}=\colvec{a\\b\\c} \end{equation*}

has no solution or explain why no such vector exists.

Question 2.16.

Give an example of a vector \(\colvec{a\\b\\c}\) such that the equation

\begin{equation*} x_1\colvec{1\\1\\0}+x_2 \colvec{0\\0\\1}=\colvec{a\\b\\c} \end{equation*}

has exactly 1 solution or explain why no such vector exists.

Question 2.17.

Give an example of a vector \(\colvec{a\\b\\c}\) such that the equation

\begin{equation*} x_1\colvec{1\\1\\0}+x_2 \colvec{0\\0\\1}+x_3 \colvec{1\\1\\1}=\colvec{a\\b\\c} \end{equation*}

has exactly 1 solution or explain why no such vector exists.

Question 2.18.

Give an example of a vector \(\colvec{a\\b\\c}\) such that the equation

\begin{equation*} x_1\colvec{1\\1\\0}+x_2 \colvec{0\\0\\1}+x_3 \colvec{1\\1\\1}=\colvec{a\\b\\c} \end{equation*}

has no solutions or explain why no such vector exists.

Question 2.19.

Give an example of a vector \(\colvec{a\\b\\c}\) such that the equation

\begin{equation*} x_1\colvec{1\\1\\0}+x_2 \colvec{0\\0\\1}+x_3 \colvec{a\\b\\c}=\colvec{7\\0\\-2} \end{equation*}

has exactly 1 solution or explain why no such vector exists.

Definition 2.10.

A linear combination of a set \(S\) is a vector of the form

\begin{equation*} \sum_{i=1}^n c_i \vec{v_i} = c_1 \vec{v_1} + c_2 \vec{v_2}+...+c_k \vec{v_k} \end{equation*}

where \(\vec{v}_i \in S\) and \(c_i \in \mathbb{R}\text{.}\) Note that

\begin{equation*} \sum_{i=1}^n c_i \vec{v_i} \end{equation*}

will not usually be in \(S\) even though \(\vec{v}_i \in S\text{.}\)

Question 2.20.

Can you write \(\vec{b}=\colvec{2\\2\\4}\) as a linear combination of \(\vec{v_1}=\colvec{2\\1\\1}\) and \(\vec{v_2}=\colvec{2\\-1\\1}\text{?}\) Justify your answer.

Question 2.21.

Repeat the previous problem for \(\vec{b}=\colvec{2\\0\\0}\) and \(\vec{b}=\colvec{2\\3\\-1}\text{.}\)

Question 2.22.

Can you write \(\vec{b}=\colvec{2\\0\\4}\) as a linear combination of \(\vec{v_1}=\colvec{1\\1\\1}\) and \(\vec{v_2}=\colvec{-1\\1\\1}\text{?}\) Justify your answer.

Definition 2.11.

We define a matrix-vector product as follows: If \(A\) is a \(m\) by \(n\) matrix,

\begin{equation*} A=\begin{bmatrix} a_{11} \amp a_{12} \amp ... \amp a_{1n} \\ a_{21}\amp a_{22}\amp ... \amp a_{2n} \\ \vdots \amp \vdots \amp \amp \vdots \\ a_{m1}\amp a_{m2} \amp ... \amp a_{mn} \end{bmatrix} \end{equation*}

and \(\vec{x}=\colvec{x_1\\x_2\\\vdots\\x_n} \in \mathbb{R}^n\text{,}\) then the matrix-vector product is given by

\begin{equation*} A\vec{x} = x_1 \colvec{a_{11}\\a_{21}\\\vdots\\a_{m1}}+x_2 \colvec{a_{12}\\a_{22}\\\vdots\\a_{m2}}+ \cdots x_n \colvec{a_{1n}\\a_{2n}\\\vdots\\a_{mn}} \end{equation*}

\(\mathbb{R}^m\text{.}\)

Question 2.23.

If \(A\) is a \(m\) by \(n\) matrix, then \(A\vec{x} \in \mathbb{R}^\diamondsuit\) for what value of \(\diamondsuit\text{?}\)

It should not surprise you that you can multiply a scalar multiple of a vector by a matrix by factoring out the scalar. In mathematical notation, \(A (k \vec{v}) = k (A\vec{v})\text{.}\) Additionally, you can apply the scalar multiplication to the matrix. In other words, \(A (k \vec{v}) = k (A\vec{v}) = (kA)\vec{v}\text{.}\) These kind of manipulations will be discussed more when we work with matrix operations later, but you may find these facts useful in your work right now. You should take time to write out the details of any of these arithmetic ideas that you think would be useful in your work.

Question 2.24.

(a)

Write out the \(k\)-th component of the resulting vector of the product

\begin{equation*} \begin{bmatrix} a_{11} \amp a_{12} \amp ... \amp a_{1n} \\ a_{21}\amp a_{22}\amp ... \amp a_{2n} \\ \vdots \amp \vdots \amp \amp \vdots \\ a_{m1}\amp a_{m2} \amp ... \amp a_{mn} \end{bmatrix} \colvec{x_1\\x_2\\\vdots\\x_n} \end{equation*}

Solution.

\(a_{k1}x_1+a_{k2}x_2+\cdots + a_{kn}x_n\)

(b)

How can you express the result of the matrix-vector product in terms of \(\vec{x}\) and the rows of \(A\text{?}\)

Solution.

The \(k\)-th component of the matrix-vector product is the dot product of row \(k\) of \(A\) with \(\vec{x}\text{.}\)

(c)

How can you express the result of the matrix-vector product in terms of \(\vec{x}\) and the columns of \(A\text{?}\)

Solution.

One way to view this is as a linear combination of the columns of \(A\) with the coefficient on the \(k\)-th column of \(A\) being \(x_k\text{.}\)

Based on the above definition of the matrix vector product, if

\begin{equation*} A=\begin{bmatrix} a_{11} \amp a_{12} \amp ... \amp a_{1n} \\ a_{21}\amp a_{22}\amp ... \amp a_{2n} \\ \vdots \amp \vdots \amp \amp \vdots \\ a_{m1}\amp a_{m2} \amp ... \amp a_{mn} \end{bmatrix} \end{equation*}

and \(\vec{b}= \colvec{b_1\\b_2\\\vdots\\b_m}\text{,}\) then by Question 2.13, \(A\vec{x} = \vec{b}\) has the same solution set as the system

\begin{equation*} \begin{array}{rcrcrcrcr} a_{11} x_1 \amp +\amp a_{12} x_2 \amp +\amp ... \amp +\amp a_{1n}x_n \amp =\amp b_1 \\ % -1 3 a_{21} x_1 \amp +\amp a_{22} x_2 \amp +\amp ... \amp +\amp a_{2n}x_n \amp =\amp b_2 \\ \vdots \amp \amp \vdots \amp \amp \amp \amp \vdots \amp \amp \vdots \\ a_{m1} x_1 \amp +\amp a_{m2} x_2 \amp +\amp ... \amp +\amp a_{mn}x_n \amp =\amp b_m \end{array} \end{equation*}

Question 2.25.

Write each of the following as a matrix equation, a vector equation, and system of equations. You need to write out the exact corresponding vector equation, matrix equation, and system of equations, not some equivalent form.

(a)

\(\begin{bmatrix} 1 \amp 2\amp 3\\4\amp 5\amp 6\\7\amp 8\amp 9 \end{bmatrix} \colvec{x_1\\x_2\\x_3} =\colvec{a\\b\\c}\)

(b)

\(a_1 \colvec{1\\2\\5}+\colvec{3\\0\\-1}+a_2 \colvec{-1\\0\\2} =\colvec{4\\5\\0}\)

(c)

\begin{align*} 2x_1+3x_2\phantom{+3x_3}\amp =7\\ -x_1+x_2+4x_3\amp =0\\ 5x_1-6x_2-x_3\amp =2 \end{align*}