The span of a set of vectors

In this exercise, we will consider the span of some sets of two- and three-dimensional vectors.

1. Consider the vectors .

   1. Is $\bvec = \twovec{2}{1}$ in $\laspan{\vvec_1,\vvec_2}$?

   2. Give a geometric description of $\laspan{\vvec_1,\vvec_2}$.

2. Consider the vectors .

   1. Is the vector $\bvec=\threevec{-10}{-1}{5}$ in $\laspan{\vvec_1,\vvec_2,\vvec_3}$?

   2. Is the vector $\vvec_3$ in $\laspan{\vvec_1,\vvec_2,\vvec_3}$?

   3. Is the vector $\bvec=\threevec{3}{3}{-1}$ in $\laspan{\vvec_1,\vvec_2,\vvec_3}$?

   4. Give a geometric description of $\laspan{\vvec_1,\vvec_2,\vvec_3}$.

Provide a justification for your response to the following questions.

1. Suppose you have a set of vectors $\vvec_1,\vvec_2,\ldots,\vvec_n$. Can you guarantee that $\zerovec$ is in $\laspan{\vvec_1\,\vvec_2,\ldots,\vvec_n}$?

2. Suppose that $A$ is an $m \times n$ matrix. Can you guarantee that the equation $A\xvec = \zerovec$ is consistent?

3. What is $\laspan{\zerovec,\zerovec,\ldots,\zerovec}$?

For both parts of this exercise, give a geometric description of sets of the vectors $\bvec$ and include a sketch.

1. For which vectors $\bvec$ in $\real^2$ is the equation consistent?

2. For which vectors $\bvec$ in $\real^2$ is the equation consistent?

Consider the following matrices: . Do the columns of $A$ span $\real^4$? Do the columns of $B$ span $\real^4$?

Determine whether the following statements are true or false and provide a justification for your response. Throughout, we will assume that the matrix $A$ has columns $\vvec_1,\vvec_2,\ldots,\vvec_n$; that is, .

1. If the equation $A\xvec = \bvec$ is consistent, then $\bvec$ is in $\laspan{\vvec_1,\vvec_2,\ldots,\vvec_n}$.

2. The equation $A\xvec = \vvec_1$ is consistent.

3. If $\vvec_1$, $\vvec_2$, $\vvec_3$, and $\vvec_4$ are vectors in $\real^3$, then their span is $\real^3$.

4. If $\bvec$ is a linear combination of $\vvec_1, \vvec_2,\ldots,\vvec_n$, then $\bvec$ is in $\laspan{\vvec_1,\vvec_2,\ldots,\vvec_n}$.

5. If $A$ is an $8032\times 427$ matrix, then the span of…

This exercise asks you to construct some matrices whose columns span a given set.

1. Construct a $3\times3$ matrix whose columns span $\real^3$.

2. Construct a $3\times3$ matrix whose columns span a plane in $\real^3$.

3. Construct a $3\times3$ matrix whose columns span a line in $\real^3$.

Provide a justification for your response to the following questions.

1. Suppose that we have vectors in $\real^8$, $\vvec_1,\vvec_2,\ldots,\vvec_{10}$, whose span is $\real^8$. Can every vector $\bvec$ in $\real^8$ be written as a linear combination of $\vvec_1,\vvec_2,\ldots,\vvec_{10}$?

2. Suppose that we have vectors in $\real^8$, $\vvec_1,\vvec_2,\ldots,\vvec_{10}$, whose span is $\real^8$. Can every vector $\bvec$ in $\real^8$ be written uniquely as a linear combination of $\vvec_1,\vvec_2,\ldots,\vvec_{10}$?

3. Do the vectors span $\real^3$?

4. Suppose that $\vvec_1,\vvec_2,\ldots,\vvec_n$ span $\real^{438}$. What can you guarantee about the value…

The following observation will be helpful in this exercise. If we want to find a solution to the equation $AB\xvec = \bvec$, we could first find a solution to the equation $A\yvec = \bvec$ and then find a solution to the equation $B\xvec = \yvec$.

Suppose that $A$ is a $3\times 4$ matrix whose columns span $\real^3$ and $B$ is a $4\times 5$ matrix. In this case, we can form the product $AB$.

1. What is the shape of the product $AB$?

2. Can you guarantee that the columns of $AB$ span $\real^3$?

3. If you…

Suppose that $A$ is a $12\times12$ matrix and that, for some vector $\bvec$, the equation $A\xvec=\bvec$ has a unique solution.

1. What can you say about the pivot positions of $A$?

2. What can you say about the span of the columns of $A$?

3. If $\cvec$ is some other vector in $\real^{12}$, what can you conclude about the equation $A\xvec = \cvec$?

4. What can you say about the solution space to the equation $A\xvec =\zerovec$?

Suppose that .

1. Is $\vvec_3$ a linear combination of $\vvec_1$ and $\vvec_2$? If so, find weights such that $\vvec_3 = a\vvec_1+b\vvec_2$.

2. Show that a linear combination can be rewritten as a linear combination of $\vvec_1$ and $\vvec_2$.

3. Explain why $\laspan{\vvec_1,\vvec_2,\vvec_3} = \laspan{\vvec_1,\vvec_2}$.

As defined in this section, the span of a set of vectors is generated by taking all possible linear combinations of those vectors. This exercise will demonstrate the fact that the span can also be realized as the solution space to a linear system.

We will consider the vectors

1. Is every vector in $\real^3$ in $\laspan{\vvec_1,\vvec_2,\vvec_3}$? If not, describe the span.

2. To describe $\laspan{\vvec_1,\vvec_2,\vvec_3}$ as the solution space of a linear system, we will write .…