# TBIL Activities for Understanding Linear Algebra

## Section3.5Subspaces of $$\real^p$$

### Activity3.5.0.1.

Let's consider the following matrix $$A$$ and its reduced row echelon form.
\begin{equation*} A = \left[\begin{array}{rrrr} 2 \amp -1 \amp 2 \amp 3 \\ 1 \amp 0 \amp 0 \amp 2 \\ -2 \amp 2 \amp -4 \amp -2 \\ \end{array}\right] \sim \left[\begin{array}{rrrr} 1 \amp 0 \amp 0 \amp 2 \\ 0 \amp 1 \amp -2 \amp 1 \\ 0 \amp 0 \amp 0 \amp 0 \\ \end{array}\right]\text{.} \end{equation*}

#### (a)

Are the columns of $$A$$ linearly independent? Do they span $$\real^3\text{?}$$

#### (b)

Give a parametric description of the solution space to the homogeneous equation $$A\xvec = \zerovec\text{.}$$

#### (c)

Explain how this parametric description produces two vectors $$\wvec_1$$ and $$\wvec_2$$ whose span is the solution space to the equation $$A\xvec = \zerovec\text{.}$$

#### (d)

What can you say about the linear independence of the set of vectors $$\wvec_1$$ and $$\wvec_2\text{?}$$

#### (e)

Let's denote the columns of $$A$$ as $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ $$\vvec_3\text{,}$$ and $$\vvec_4\text{.}$$ Explain why $$\vvec_3$$ and $$\vvec_4$$ can be written as linear combinations of $$\vvec_1$$ and $$\vvec_2\text{.}$$

#### (f)

Explain why $$\vvec_1$$ and $$\vvec_2$$ are linearly independent and $$\laspan{\vvec_1,\vvec_2} = \laspan{\vvec_1, \vvec_2, \vvec_3, \vvec_4}\text{.}$$

### Definition3.5.0.1.

A subspace of $$\real^p$$ is a nonempty subset of $$\real^p$$ such that any linear combination of vectors in that set is also in the set.

### Activity3.5.0.2.

We will look at some subspaces of $$\real^2\text{.}$$

#### (a)

Explain why a line that does not pass through the origin is not a subspace of $$\real^2\text{.}$$

#### (b)

Explain why any subspace of $$\real^2$$ must contain the zero vector $$\zerovec\text{.}$$

#### (c)

Explain why the subset $$S$$ of $$\real^2$$ that consists of only the zero vector $$\zerovec$$ is a subspace of $$\real^2\text{.}$$

#### (d)

Explain why the subspace $$S=\real^2$$ is itself a subspace of $$\real^2\text{.}$$

#### (e)

If $$\vvec$$ and $$\wvec$$ are two vectors in a subspace $$S\text{,}$$ explain why $$\laspan{\vvec,\wvec}$$ is contained in the subspace $$S$$ as well.

### Definition3.5.0.2.

A basis for a subspace $$S$$ of $$\real^p$$ is a set of vectors in $$S$$ that are linearly independent and span $$S\text{.}$$ It can be seen that any two bases have the same number of vectors. Therefore, we say that the dimension of the subspace $$S\text{,}$$ denoted $$\dim S\text{,}$$ is the number of vectors in any basis.

### Definition3.5.0.3.

If $$A$$ is an $$m\times n$$ matrix, we call the subset of vectors $$\xvec$$ in $$\real^n$$ satisfying $$A\xvec = \zerovec$$ the null space of $$A\text{.}$$ We denote it as $$\nul(A)\text{.}$$
We will explore some null spaces in the next couple of activities.

### Activity3.5.0.3.

Consider the matrix
\begin{equation*} A=\left[\begin{array}{rrr} 1 \amp 3 \amp -1 \\ -2 \amp 0 \amp -4 \\ 1 \amp 2 \amp 0 \\ \end{array}\right] \end{equation*}

#### (a)

Give a parametric description of the null space $$\nul(A)\text{.}$$

#### (b)

Give a basis for and state the dimension of $$\nul(A)\text{.}$$

#### (c)

The null space $$\nul(A)$$ is a subspace of $$\real^p$$ for which $$p\text{?}$$

### Activity3.5.0.4.

Consider the matrix $$A$$ whose reduced row echelon form is given:
\begin{equation*} A \sim \left[\begin{array}{rrrr} 1 \amp 2 \amp 0 \amp -3 \\ 0 \amp 0 \amp 1 \amp 2 \\ \end{array}\right]\text{.} \end{equation*}

#### (a)

Give a parametric description of $$\nul(A)\text{.}$$

#### (b)

Notice that the parametric description gives a set of vectors that span $$\nul(A)\text{.}$$ Explain why this set of vectors is linearly independent and hence forms a basis. What is the dimension of $$\nul(A)\text{?}$$

#### (c)

For this matrix, $$\nul(A)$$ is a subspace of $$\real^p$$ for what $$p\text{?}$$

### Activity3.5.0.5.

#### (a)

What is the relationship between the dimensions of the matrix $$A\text{,}$$ the number of pivot positions of $$A$$ and the dimension of $$\nul(A)\text{?}$$

#### (b)

Suppose that the columns of a matrix $$A$$ are linearly independent. What can you say about $$\nul(A)\text{?}$$

#### (c)

If $$A$$ is an invertible $$n\times n$$ matrix, what can you say about $$\nul(A)\text{?}$$

### Activity3.5.0.6.

Suppose that $$A$$ is a $$5\times 10$$ matrix and that $$\nul(A) = \real^{10}\text{.}$$ What can you say about the matrix $$A\text{?}$$

### Definition3.5.0.4.

The rank of a matrix $$A\text{,}$$ denoted $$\rank(A)\text{,}$$ is the number of pivot positions of $$A\text{.}$$

### Definition3.5.0.5.

If $$A$$ is an $$m\times n$$ matrix, we call the span of its columns the column space of $$A$$ and denote it as $$\col(A)\text{.}$$
We will explore some column spaces in the next couple of activities.

### Activity3.5.0.7.

Consider the matrix
\begin{equation*} A= \left[\begin{array}{rrr} \vvec_1 \amp \vvec_2 \amp \vvec_3 \end{array}\right] = \left[\begin{array}{rrr} 1 \amp 3 \amp -1 \\ -2 \amp 0 \amp -4 \\ 1 \amp 2 \amp 0 \\ \end{array}\right]\text{.} \end{equation*}

#### (a)

Since $$\col(A)$$ is the span of the columns, the vectors $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3$$ naturally span $$\col(A)\text{.}$$ Are these vectors linearly independent?

#### (b)

Show that $$\vvec_3$$ can be written as a linear combination of $$\vvec_1$$ and $$\vvec_2$$ by giving the linear combination. Then explain why $$\col(A)=\laspan{\vvec_1,\vvec_2}\text{.}$$

#### (c)

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

### Activity3.5.0.8.

Consider the matrix $$A$$ and its reduced row echelon form:
\begin{equation*} A = \left[\begin{array}{rrrr} -2 \amp -4 \amp 0 \amp 6 \\ 1 \amp 2 \amp 0 \amp -3 \\ \end{array}\right] \sim \left[\begin{array}{rrrr} 1 \amp 2 \amp 0 \amp -3 \\ 0 \amp 0 \amp 0 \amp 0 \\ \end{array}\right]\text{.} \end{equation*}
We will call the columns $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ $$\vvec_3\text{,}$$ and $$\vvec_4\text{.}$$

#### (a)

Explain why $$\vvec_2\text{,}$$ $$\vvec_3\text{,}$$ and $$\vvec_4$$ can be written as a linear combination of $$\vvec_1\text{.}$$

#### (b)

Explain why $$\col(A)$$ is a 1-dimensional subspace of $$\real^2$$ and is therefore a line.

#### (c)

What is the relationship between the dimension $$\dim~\col(A)$$ and the rank $$\rank(A)\text{?}$$

#### (d)

What is the relationship between the dimension of the column space $$\col(A)$$ and the null space $$\nul(A)\text{?}$$

### Activity3.5.0.9.

If $$A$$ is an invertible $$9\times9$$ matrix, what can you say about the column space $$\col(A)\text{?}$$

### Activity3.5.0.10.

If $$\col(A)=\{\zerovec\}\text{,}$$ what can you say about the matrix $$A\text{?}$$

### Activity3.5.0.11.

Consider a matrix $$A\text{.}$$ Which of the following is equal to the dimension of the null space of $$A\text{?}$$
1. The number of pivot columns
2. The number of non-pivot columns
3. The number of pivot rows
4. The number of non-pivot rows

### Activity3.5.0.12.

Consider a matrix $$A\text{.}$$ Which of the following is equal to the dimension of the column space of $$A\text{?}$$
1. The number of pivot columns
2. The number of non-pivot columns
3. The number of pivot rows
4. The number of non-pivot rows

### Activity3.5.0.13.

Consider the matrix
\begin{equation*} A = \left[\begin{array}{ccc} 1 & -3 & 2\\ 2 & -6 & 0 \\ 0 & 0 & 1 \\ -1 & 3 & 1 \end{array}\right] . \end{equation*}
Verify that if $$A$$ is an $$m\times n$$ matrix, then
\begin{equation*} \dim~\nul(A) + \dim~\col(A) = n \end{equation*}
holds for $$A\text{.}$$ This equation is also known as the Rank-Nullity Theorem.