91Ó°ÊÓ

In Exercises \(57-62,\) determine whether \(\mathbf{b}\) is in the column space of \(A\). If it is, write \(b\) as a linear combination of the column vectors of \(A\). $$A=\left[\begin{array}{rrr}-1 & -1 & 1 \\\1 & 0 & 1 \\\\-3 & -2 & 1\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{r}0 \\\3 \\\\-3\end{array}\right]$$

Prove that the nonzero row vectors of a matrix in row-echelon form are linearly independent.

Show that the three points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) an a plane are collinear if and only if the matrix. \(\left[\begin{array}{lll}x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{array}\right]\) has rank less than 3 .

Perform a rotation of axes to eliminate the xy-term, and sketch the graph of the conic. $$ 5 x^{2}-6 x y+5 y^{2}-12=0 $$

Let \(A\) be an \(m \times n\) matrix. Prove that \(N(A) \subset N\left(A^{T} A\right)\).

CAPSTONE The dimension of the row space of a \(3 \times 5\) matrix \(A\) is 2 (a) What is the dimension of the column space of \(A ?\) (b) What is the rank of \(A ?\) (c) What is the nullity of \(A ?\) (d) What is the dimension of the solution space of the homogencous system \(A x=0 ?\)

Prove that row operations do not change the dependency relationships among the columns of an \(m \times n\) matrix.

Proof Prove that row operations do not change the dependency relationships among the columns of am \(m \times n\) matrix.