91Ó°ÊÓ

In Exercises 19 through \(24,\) find the matrix \(B\) of the linear transformation \(T(\vec{x})=A \vec{x}\) with respect to the basis \(\mathfrak{B}=\left(\vec{v}_{1}, \vec{v}_{2}\right) .\) For practice, solve each problem in three ways: (a) Use the formula \(B=S^{-1} A S,(b)\) use a commutative diagram (as in Examples 3 and 4), and (c) construct \(\boldsymbol{B}^{*}\)column by column." $$A=\left[\begin{array}{rr} 13 & -20 \\ 6 & -9 \end{array}\right] ; \vec{v}_{1}=\left[\begin{array}{l} 2 \\ 1 \end{array}\right], \vec{v}_{2}=\left[\begin{array}{l} 5 \\ 3 \end{array}\right]$$

Find a redundant column vector of the given matrix \(A\), and write it as a linear combination of preceding columns. Use this representation to write a nontrivial relation among the columns, and thus find a nonzero vector in the kernel of A. (This procedure is illustrated in Example \(8 .\)) $$\left[\begin{array}{lll} 1 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{array}\right]$$

Find a redundant column vector of the given matrix \(A\), and write it as a linear combination of preceding columns. Use this representation to write a nontrivial relation among the columns, and thus find a nonzero vector in the kernel of A. (This procedure is illustrated in Example \(8 .\)) $$\left[\begin{array}{lll} 1 & 3 & 6 \\ 1 & 2 & 5 \\ 1 & 1 & 4 \end{array}\right]$$

Find a basis of the subspace of \(\mathbb{R}^{3}\) defined by the equation \\[ 2 x_{1}+3 x_{2}+x_{3}=0 \\]

Give an example of a linear transformation whose image is the line spanned by \\[ \left[\begin{array}{l} 7 \\ 6 \\ 5 \end{array}\right] \\] in \(\mathbb{R}^{3}\)

Give an example of a linear transformation whose kernel is the plane \(x+2 y+3 z=0\) in \(\mathbb{R}^{3}\)

Give an example of a linear transformation whose kernel is the line spanned by \\[ \left[\begin{array}{r} -1 \\ 1 \\ 2 \end{array}\right] \\] in \(\mathbb{R}^{3}\)

Consider a nonzero vector \(\vec{v}\) in \(\mathbb{R}^{n} .\) What is the dimension of the space of all vectors in \(\mathbb{R}^{n}\) that are perpendicular to \(\vec{v} ?\)

Let \(\mathfrak{B}=\left(\vec{v}_{1}, \vec{v}_{2}, \vec{v}_{3}\right)\) be any basis of \(\mathbb{R}^{3}\) consisting of per pendicular unit vectors, such that \(\vec{v}_{3}=\vec{v}_{1} \times \vec{v}_{2} .\) In Ex ercises 31 through \(36,\) find the \(\$$ -matrix \)B\( of the given linear transformation \)T\( from \)\mathbb{R}^{3}\( to \)\mathbb{R}^{3}\(. Interpret \)\boldsymbol{T}$ geometrically. $$T(\vec{x})=\vec{v}_{1} \times \vec{x}+\left(\vec{v}_{1} \cdot \vec{x}\right) \vec{v}_{1}$$

Consider a square matrix \(A\) a. What is the relationship among ker(A) and \(\operatorname{ker}\left(A^{2}\right) ?\) Are they necessarily equal? Is one of them necessarily contained in the other? More generally, what can you say about \(\operatorname{ker}(A), \operatorname{ker}\left(A^{2}\right), \operatorname{ker}\left(A^{3}\right), \ldots ?\) b. What can you say about im(A), im(A^), \(\operatorname{im}\left(A^{3}\right), \ldots ?\)