SAMPLE EXAM IN ADVANCED LINEAR ALGEBRA

Do as many as you can. Write down all details for partial credits.

1. State the definition of "vector space over a field".

2. Let $V$ be a vector space over a field $F$ and let $\alpha, \beta$ and $\gamma$ be vectors for which $\alpha+\beta=\alpha+\gamma$. Show that $\beta=\gamma$.

3. Let $V$ be a vector space over a field $F$. Show that $0\cdot \alpha=0$ for every vector $\alpha$.

4. What does it mean to say that a finite set of vectors is "linearly independent over the field $F$"?

5. Let $\alpha_1(t)=e^t, \ \alpha_2(t)=e^{2t},\ \alpha_3(t)=e^{3t},\ \alpha_4(t)=e^{4t}$. Show that these four functions are linearly independent over the reals.

6. What is meant by a "basis" for a subspace $W$ of a vector space $V$?

7. Let $V={\bf R}^{4\times 1}$ and

$$\alpha_1= \begin{pmatrix} 1\\ 2\\ 1\\ 1 \end{pmatrix}, \ \alp... ...2 \end{pmatrix}, \ \alpha_4= \begin{pmatrix} 0\\ 2\\ 0\\ 0 \end{pmatrix}.$$

Find a basis for the span of $\alpha_1, \alpha_2, \alpha_3$ and $\alpha_4$.

8. Let

$$A = \begin{pmatrix} 1 & 1 & 2 & 0\\ 2 & 0 & 2 & 2 \\ 1 & 1 & 2 & 0\\ 1 & 1 & 2 & 0\\ \end{pmatrix}$$

be a matrix over the reals.

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

(b) Determine the nullity of $A$.

9. Let

$$A = \begin{pmatrix} 1 & 1 & 3& 1 \\ 2 & 1 & 1 & 2 \\ 3 & 1 & 4 & 1\\ 4 & 1 & 1 & 1\\ \end{pmatrix},$$

considered as a matrix over the reals. Evaluate $det \ A$.

10. In the following system of linear equations, use Cramer's Rule to find the value of $y$:

$$\left\{ \begin{array}{ll} & x+y+3z+w=1,\\ & 2x+y+z+2w=1,\\ & 3x+y+4z+w=1,\\ & 4x+y+z+w=1. \end{array}\right.$$ (1)

11. Let

$$A = \begin{pmatrix} 1 & 1 & 2 & 0\\ 1 & 0 & 1 & 1 \\ 1 & 1 & 2 & 0\\ 1 & 0 & 1 & 1\\ \end{pmatrix},$$

considered as a matrix over the reals.

(a) Find a linear homogeneous system of equations that charactersizes the column space of $A$.

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

12. Let

$$A = \begin{pmatrix} 1 & 2 & 1 & 1\\ 1 & 1 & 1 & 2 \\ 2 & 3 & 2 & 2\\ 0 & 1 & 0 & 4\\ \end{pmatrix},$$

considered as a matrix over the reals.

Find the row-reduced echelon form $R$ of $A$ and an invertible matrix $P$ such that $PA=R$.

13. Let

$$\alpha_1= \begin{pmatrix} 1\\ 0\\ 0\\ 0 \end{pmatrix}, \ \alp... ... 1\\ 1 \end{pmatrix}, \ and\ B=\{\alpha_1, \alpha_2, \alpha_3, \alpha_4\}.$$

Let

$$\beta_1= \begin{pmatrix} 1\\ 1\\ 0\\ 0 \end{pmatrix}, \ \beta... ...0\\ 0\\ 0 \end{pmatrix}, \ and\ B'=\{\beta_1, \beta_2, \beta_3, \beta_4\}.$$

(a) Find the matrix which converts the coordinates of a vector relative to the basis $B$ into coordinates of the same vector relative to $B'$. Both are bases for ${\bf R}^4$.

(b) Find the matrix which converts the coordinates of a vector relative to the basis $B'$ into coordinates of the same vector relative to $B$.

14. State what is meant by a "linear transformation".

15. Let $V$ and $W$ be vector spaces over the field $F$ and $T: V\rightarrow W$ a linear transformation. Finally, let $U$ be a subspace of $W$. Define the subset $M$ of $V$ as follows:

$$M=\{\alpha:\ \alpha\in V\ and\ T(\alpha)\in U\}.$$

Show that $M$ is a subspace of $V$.

16. Let

$$A = \begin{pmatrix} 1 & 2 & 1 \\ 1 & 1 & 2 \\ 2 & 3 & 3\\ \end... ...atrix} 1\\ 1\\ 1\\ \end{pmatrix},\ and\ B=\{\alpha_1, \alpha_2, \alpha_3\}.$$

Find the rule, in standard coordinates, for the linear operator $T: {\bf R}^{3\times 1} \rightarrow {\bf R}^{3 \times 1}$ such that $A$ is the matrix of $T$ relative to the basis $B$.

17. Let $T$ be the function on ${\bf R}^{3\times 1}$ whose rule, in standard coordinates is given by

$$T \begin{pmatrix} x\\ y\\ z\\ \end{pmatrix}= \begin{pmatrix} x+y\\ y+z\\ x+z\\ \end{pmatrix}.$$

(a) Show that $T$ is a linear operator on ${\bf R}^{3\times 1}$.

(b) Let

$$\alpha_1= \begin{pmatrix} 1\\ 0\\ 0\\ \end{pmatrix}, \ \alpha... ...atrix} 1\\ 1\\ 1\\ \end{pmatrix}, \ and\ B=\{\alpha_1, \alpha_2, \alpha_3\}$$

Find the matrix $[T]_B$ of $T$ relative to the basis $B$.

(c) Let

$$\beta_1= \begin{pmatrix} 1\\ 1\\ 0\\ \end{pmatrix}, \ \beta_2... ...pmatrix} 1\\ 0\\ 0\\ \end{pmatrix}, \ and\ B'=\{\beta_1, \beta_2, \beta_3\}$$

Find the matrix $[T]_B$ of $T$ relative to the basis $B'$.

(d) Find an invertible matrix $Q$ such that $[T]_{B'}=Q=[T]_B Q^{-1}$.

(e) Show that $T$ is invertible and give the rule for $T^{-1}$ in standard coordinates.