Tagged: column vector

Problem 708

Let $A=\begin{bmatrix} 2 & 4 & 6 & 8 \\ 1 &3 & 0 & 5 \\ 1 & 1 & 6 & 3 \end{bmatrix}$.

(a) Find a basis for the nullspace of $A$.

(b) Find a basis for the row space of $A$.

(c) Find a basis for the range of $A$ that consists of column vectors of $A$.

(d) For each column vector which is not a basis vector that you obtained in part (c), express it as a linear combination of the basis vectors for the range of $A$.

Problem 659

Fix the row vector $\mathbf{b} = \begin{bmatrix} -1 & 3 & -1 \end{bmatrix}$, and let $\R^3$ be the vector space of $3 \times 1$ column vectors. Define
$W = \{ \mathbf{v} \in \R^3 \mid \mathbf{b} \mathbf{v} = 0 \}.$ Prove that $W$ is a vector subspace of $\R^3$.

Problem 656

Suppose that an $n \times m$ matrix $M$ is composed of the column vectors $\mathbf{b}_1 , \cdots , \mathbf{b}_m$.

Prove that a vector $\mathbf{v} \in \R^n$ can be written as a linear combination of the column vectors if and only if there is a vector $\mathbf{x}$ which solves the equation $M \mathbf{x} = \mathbf{v}$.

Problem 654

Suppose $M$ is an $n \times n$ upper-triangular matrix.

If the diagonal entries of $M$ are all non-zero, then prove that the column vectors are linearly independent.

Does the conclusion hold if we do not assume that $M$ has non-zero diagonal entries?

Problem 580

(a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly dependent.

(b) Prove that the row vectors of every $5\times 3$ matrix $B$ are linearly dependent.

Problem 136

Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.