91Ó°ÊÓ

a. Let \(V\) be a subspace of \(\mathbb{R}^{n}\). Let \(m\) be the largest number of linearly independent vectors we can find in \(V\). (Note that \(m \leq n\), by Theorem 3.2 .8 ) Choose linearly independent vectors \(\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{m}\) in \(V\) Show that the vectors \(\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{m}\) span \(V\) and are therefore a basis of \(V\). This exercise shows that any subspace of \(\mathbb{R}^{n}\) has a basis. If you are puzzled, think first about the special case when \(V\) is a plane in \(\mathbb{R}^{3}\). What is \(m\) in this case? b. Show that any subspace \(V\) of \(\mathbb{R}^{n}\) can be represented as the image of a matrix.

Consider an \(n \times p\) matrix \(A\) and a \(p \times m\) matrix \(B\) a. What is the relationship between \(\operatorname{ker}(A B)\) and \(\operatorname{ker}(B) ?\) Are they always equal? Is one of them always contained in the other? b. What is the relationship between im(A) and \(\operatorname{im}(A B) ?\)

Consider an \(m \times n\) matrix \(A\) and an \(n \times m\) matrix \(B\) (with \(n \neq m\) ) such that \(A B=I_{m}\). (We say that \(A\) is a left inverse of \(B\).) Are the columns of \(B\) linearly independent? What about the columns of \(A\) ?

Consider linearly independent vectors \(\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{m}\) in \(\mathbb{R}^{n},\) and let \(A\) be an invertible \(m \times m\) matrix. Are the columns of the following matrix linearly independent? \\[ \left[\begin{array}{cccc} | & | & & | \\ \vec{v}_{1} & \vec{v}_{2} & \dots & \vec{v}_{m} \\ | & | & & | \end{array}\right] A \\]

Consider a matrix \(A,\) and let \(B=\operatorname{rref}(A)\) a. Is ker(A) necessarily equal to ker( \(B\) )? Explain. b. Is im \((A)\) necessarily equal to im \((B) ?\) Explain.

Are the columns of an invertible matrix linearly independent?

Find a basis of the kernel of the matrix \\[ \left[\begin{array}{ccccc} 1 & 2 & 0 & 3 & 5 \\ 0 & 0 & 1 & 4 & 6 \end{array}\right] \\]. Justify your answer carefully; that is, explain how you know that the vectors you found are linearly independent and span the kernel.

Express the plane \(V\) in \(\mathbb{R}^{3}\) with equation \(3 x_{1}+4 x_{2}+\) \(5 x_{3}=0\) as the kernel of a matrix \(A\) and as the image of a matrix \(B\).

For which values of the constants \(a, b, c, d, e,\) and \(f\) are the following vectors linearly independent? Justify your answer. $$\left[\begin{array}{l} a \\ 0 \\ 0 \\ 0 \end{array}\right], \quad\left[\begin{array}{l} b \\ c \\ 0 \\ 0 \end{array}\right], \quad\left[\begin{array}{l} d \\ e \\ f \\ 0 \end{array}\right]$$

Consider the basis \(\$$ of \)\mathbb{R}^{2}\( consisting of the vectors \)\left[\begin{array}{l}1 \\ 1\end{array}\right]\( and \)\left[\begin{array}{l}1 \\\ 2\end{array}\right],\( and let \)\Re\( be the basis consisting of \)\left[\begin{array}{l}1 \\ 2\end{array}\right]\( \)\left[\begin{array}{l}3 \\ 4\end{array}\right] .\( Find a matrix \)P\( such that \\[ [\vec{x}]_{\Re}=P[\vec{x}]_{\mathfrak{B}} \\] for all \)\vec{x}\( in \)\mathbb{R}^{2}.$