Subspaces

We will explore some column spaces in this activity.

1. Consider the matrix Since $\col(A)$ is the span of the columns, we have Explain why $\vvec_3$ can be written as a linear combination of $\vvec_1$ and $\vvec_2$ and why $\col(A)=\laspan{\vvec_1,\vvec_2}$.

2. Explain why the vectors $\vvec_1$ and $\vvec_2$ form a basis for $\col(A)$ and why $\col(A)$ is a 2-dimensional subspace of $\real^3$ and therefore a plane.

3. Now consider the matrix $B$ and its reduced row echelon form: Explain why $\col(B)$ is a 1-dimensiona…

We will explore some null spaces in this activity and see why $\nul(A)$ satisfies the definition of a subspace.

1. Consider the matrix and give a parametric description of the solution space to the equation $A\xvec = \zerovec$. In other words, give a parametric description of $\nul(A)$.

2. This parametric description shows that the vectors satisfying the equation $A\xvec=\zerovec$ can be written as a linear combination of a set of vectors. In other word…

Suppose that $A$ and its reduced row echelon form are .

1. The null space $\nul(A)$ is a subspace of $\real^p$ for what $p$? The column space $\col(A)$ is a subspace of $\real^p$ for what $p$?

2. What are the dimensions $\dim~\nul(A)$ and $\dim~\col(A)$?

3. Find a basis for the column space $\col(A)$.

4. Find a basis for the null space $\nul(A)$.

Suppose that .

1. Is the vector $\threevec{0}{-1}{-1}$ in $\col(A)$?

2. Is the vector $\fourvec{2}{1}{0}{2}$ in $\col(A)$?

3. Is the vector $\threevec{2}{-2}{0}$ in $\nul(A)$?

4. Is the vector $\fourvec{1}{-1}{3}{-1}$ in $\nul(A)$?

5. Is the vector $\fourvec{1}{0}{1}{-1}$ in $\nul(A)$?

Determine whether the following statements are true or false and provide a justification for your response. Unless otherwise stated, assume that $A$ is an $m\times n$ matrix.

1. If $A$ is a $127\times 341$ matrix, then $\nul(A)$ is a subspace of $\real^{127}$.

2. If $\dim~\nul(A) = 0$, then the columns of $A$ are linearly independent.

3. If $\col(A) = \real^m$, then $A$ is invertible.

4. If $A$ has a pivot position in every column, then $\nul(A) = \real^n$.

5. If $\col(A) = \real^m$ and $\nul(A) = \{\zerovec\}$, then $A$ is invertible.

Explain why the following statements are true.

1. If $B$ is invertible, then $\nul(BA) = \nul(A)$.

2. If $B$ is invertible, then $\col(AB) = \col(A)$.

3. If $A\sim A'$, then $\nul(A) = \nul(A')$.

For each of the following conditions, construct a $3\times 3$ matrix having the given properties.

1. $\dim~\nul(A) = 0$.

2. $\dim~\nul(A) = 1$.

3. $\dim~\nul(A) = 2$.

4. $\dim~\nul(A) = 3$.

Suppose that $A$ is a $3\times 4$ matrix.

1. Is it possible that $\dim~\nul(A) = 0$?

2. If $\dim~\nul(A) = 1$, what can you say about $\col(A)$?

3. If $\dim~\nul(A) = 2$, what can you say about $\col(A)$?

4. If $\dim~\nul(A) = 3$, what can you say about $\col(A)$?

5. If $\dim~\nul(A) = 4$, what can you say about $\col(A)$?

Suppose we have the vectors and that $A$ is a matrix such that $\col(A)=\laspan{\vvec_1,\vvec_2}$ and $\nul(A) = \laspan{\wvec_1,\wvec_2}$.

1. What are the dimensions of $A$?

2. Find such a matrix $A$.

Suppose that $A$ is an $8\times 8$ matrix and that $\det A = 14$.

1. What can you conclude about $\nul(A)$?

2. What can you conclude about $\col(A)$?

Suppose that $A$ is a matrix and there is an invertible matrix $P$ such that .

1. What can you conclude about $\nul(A)$?

2. What can you conclude about $\col(A)$?