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7.B Week 2

In-class exercises

2.16, 2.17, 2.18(i), 2.22(i), 2.25(i) 2.28, 2.31(i)(iii), 2.35(i)(ii)(iii), 2.41(iii).

Workshop exercises

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Potential workshop test questions (solutions available after Wednesday): 2.18(ii), 2.23, 2.31(ii), 2.35(iv)(v)(vi), 2.40, 2.55, 2.56, 2.57(iii)(iv)(v), 2.58, 2.59.
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Other exercises: 2.18(iii), 2.22(ii)(iii), 2.25(ii), 2.29, 2.41(i)(ii), 2.57(i)(ii).

Weekly true / false quiz (closes at 2:00pm, Saturday 21 October 2017)

[10 marks total.]

Q1.

– Vector spaces.

The vector spaces in this question, and their operations, are all defined in Example 2.1.
Determine which of the following statements are true.
- (a)
  
  In the vector space ${\mathbb{R}}^{2}$ over the field ${\mathbb{R}}$ , an example of the addition operation is
  
  $(1,2)+(3,4)=(4,6).$
- (b)
  
  In the vector space ${\mathbb{C}}^{3}$ over the field ${\mathbb{C}}$ , an example of the scalar multiplication operation is
  
  $i(2,i,1+3i)=(2i,1,i+3).$
- (c)
  
  The polynomial $3+2\sqrt{2}x-4x^{2}$ is an element in the vector space $\mathcal{P}_{2}({\mathbb{Q}})$ over the field ${\mathbb{Q}}$ .
- (d)
  
  In the vector space $\operatorname{M}_{{2}}({{\mathbb{R}}})$ over ${\mathbb{R}}$ , the addition of matrices is defined to be $A+B:=AB$ by matrix multiplication.
Q2.

– Linear combinations

Recall that if $V$ is a vector space, a linear combination of the vectors $\mathbf{x_{1}},\cdots,\mathbf{x_{n}}$ is any vector of the form $\alpha_{1}\mathbf{x_{1}}+\cdots+\alpha_{n}\mathbf{x_{n}}$ where $\alpha_{i}\in F$ are elements of the field of $V$ .
Determine which of the following statements are true.
- (a)
  
  In the vector space ${\mathbb{R}}^{3}$ over the field ${\mathbb{R}}$ , the vector $(1,0,0)$ is a linear combination of $(1,0,1)$ and $(2,3,0)$ .
- (b)
  
  In the vector space ${\mathbb{R}}^{3}$ over the field ${\mathbb{R}}$ , the vector $(0,0,0)$ is a linear combination of $(1,0,1)$ and $(2,3,0)$ .
- (c)
  
  In the vector space ${\mathbb{C}}^{3}$ over the field ${\mathbb{C}}$ , the vector $(2i,3i,0)$ is a linear combination of $(1,0,1)$ and $(2,3,0)$ .
- (d)
  
  In the vector space ${\mathbb{C}}^{3}$ over the field ${\mathbb{C}}$ , any vector can be written as a linear combination of $(1,0,1)$ and $(2,3,0)$ .
Q3.

– Span

Recall that the span of a sequence of vectors in a vector space is the set of all possible linear combinations of that sequence.
Determine which of the following statements are true.
- (a)
  
  If $\mathbf{x},\mathbf{y}\in{\mathbb{C}}^{3}$ then $\operatorname{span}_{{\mathbb{R}}}\{\mathbf{x},\mathbf{y}\}=\operatorname{span% }_{{\mathbb{C}}}\{\mathbf{x},\mathbf{y}\}$ .
- (b)
  
  If $\mathbf{x},\mathbf{y},\mathbf{z}\in{\mathbb{R}}^{3}$ and $0\mathbf{z}\in\operatorname{span}_{{\mathbb{R}}}\{\mathbf{x},\mathbf{y}\}$ then $\mathbf{z}$ is a linear combination of $\mathbf{x},\mathbf{y}$ .
- (c)
  
  In ${\mathbb{R}}^{3}$ , the span of $(1,0,1)$ contains infinitely many elements.
- (d)
  
  If $\mathbf{x},\mathbf{y},\mathbf{z}\in{\mathbb{R}}^{3}$ is such that $\mathbf{z}=3\mathbf{x}-2\mathbf{y}$ , then $\mathbf{z}\in\operatorname{span}_{{\mathbb{R}}}\{\mathbf{x},\mathbf{y}\}$ .
Q4.

– Linear independence

Recall that a sequence of vectors $\mathbf{x_{1}},\cdots,\mathbf{x_{n}}$ in a vector space is linearly independent when it is not possible to write any of the vectors in the sequences as a linear combination of the other vectors. So also Theorem 2.20 for a more useful equivalent formulation.
Determine which of the following statements are true.
- (a)
  
  If $\mathbf{x}\in V$ is linearly independent and $\mathbf{y}\in V$ is linearly independent, then $\mathbf{x},\mathbf{y}$ is a linearly independent sequence,
- (b)
  
  If $\mathbf{x},\mathbf{y},\mathbf{z}$ is a linearly independent sequence, then $\mathbf{x},\mathbf{y}$ is linearly independent,
- (c)
  
  The sequence of three vectors $x^{3}-x,x^{2}+x,0$ , form a linearly independent sequence in the vector space $\mathcal{P}_{3}({\mathbb{R}})$ .
- (d)
  
  It is impossible to have a sequence of two vectors in ${\mathbb{R}}^{3}$ which is linearly independent.
Q5.

– Basis
For each of the following sequences of vectors, determine which ones form a basis of the subspace $W=\{\begin{bmatrix}a&b\\ 0&c\end{bmatrix}\;|\;a,b,c\in{\mathbb{R}}\}\subset\operatorname{M}_{{2}}({{% \mathbb{R}}})$ of upper-triangular 2 by 2 real matrices.
- (a)
  
  $\begin{bmatrix}1&0\\ 0&0\end{bmatrix},\begin{bmatrix}0&-1\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\ 0&1\end{bmatrix}$ is a basis of $W$ ,
- (b)
  
  $\begin{bmatrix}1&1\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\ 0&1\end{bmatrix},\begin{bmatrix}1&1\\ 0&-1\end{bmatrix}$ is a basis of $W$ ,
- (c)
  
  $\begin{bmatrix}1&0\\ 0&0\end{bmatrix},\begin{bmatrix}0&1\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\ 1&0\end{bmatrix},\begin{bmatrix}0&0\\ 0&1\end{bmatrix}$ is a basis of $W$ ,
- (d)
  
  $\begin{bmatrix}1&0\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\ 0&1\end{bmatrix}$ is a basis of $W$ .