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Exercise 1: Dimension and the Zero Vector in Different Vector Spaces

A

State the zero-vector and the dimension for the following vector spaces

  1. $\Bbb R^{4}$.

  2. $\Bbb C^{4}$.

  3. $C^{0}(\left[\,0;\,1\,\right])$.

  4. $\Bbb R^{4 \times 2}$.

  5. $P_{4}(\Bbb R)$.

Exercise 2: Linear Dependence or Independence

B

Investigate whether the following systems of vectors are linearly dependent or linearly independent. If the vectors are linearly dependent you should write one of the vectors as a linear combination of the other vectors.:

  1. By hand. $(1,2,1,0), (2,7,3,1), (3,12,5,2)$ in ${\Bbb R}^{4}$.

  2. By hand. $(1,i), (1+i,-1+i)$ in ${\Bbb C}^{2}$.

  3. $1 + 2x + 3x^{2} + x^{3} , 2 + 5x - x^{2} + x^{3} , -3 + 2x -4x^{2} -2x^{3}$ in $P_{3}(\Bbb R)$ .

  4. $\left[ \begin{array}{rrr} 1 & 2 & 0 \\ 1 & 1 & 1 \end{array}\right], \left[ \begin{array}{rrr} 1 & 1 & 2 \\ 0 & 0 & 1 \end{array}\right], \left[ \begin{array}{rrr} 2 & 5 & -2 \\ 3 & 3 & 2 \end{array}\right] $ i ${\Bbb R}^{2\times 3}$.

Exercise 3: Bases and Coordinates

C

By hand: Determine the value of $a$ that must be avoided, if the set

$$ \big ( \,(1,2,3),(-1,0,2),(1,6,a)\,\big )$$

shall be a valid base for $\mathbb R^3\,$.

D

In $\mathbb R^4$ five vectors are given: $\ma_1=(1,-1,2,1),\,\ma_2=(0,1,1,3),\,\ma_3=(1,-2,2,-1)\, \ma_4=(0,1,-1,3)$ and $\mv=(1,-2,2,-3)\,.$

Explain that $(\ma_1,\ma_2,\ma_3,\ma_4)$ is a basis for $\mathbb R^4\,$, and determine the coordinate vector $_\mathrm a\mv\,$.

Choose between the two variants of the following exercise, 4a (moderately difficult) or 4b (hard)!

Exercise 4a: Bases and Coordinates

given that the vectorspace $P_2(\mathbb R)$ has a basis $p=\big(P_1(x),P_2(x),P_3(x)\,\big)$ where

$$P_1(x)=1+x^2,\, P_2(x)=-1-x-3x^2 \,\,\,\,\mathrm{and}\,\,\,\,P_3(x)=6+x+5x^2\,$$
A

Determine the coordinate vectors for the polynomials

$$Q_1(x)=3+2x+7x^2,\; Q_2(x)=2+x+4x^2\,\,\,\,\mathrm{og}\,\,\,\,Q_3(x)=5+2x^2$$

with repsect to the basis $\,p\,.$

Exercise 4b: Bases and Coordinates (Advanced)

B

Given that the vector space $P_2(\mathbb R)$ has a basis $\big(P_1(x),P_2(x),P_3(x)\,\big)$ and that the polynomials

$$Q_1(x)=3+2x+7x^2,\; Q_2(x)=2+x+4x^2\,\,\,\,\mathrm{and}\,\,\,\,Q_3(x)=5+2x^2$$

with respect to this basis have the set of coordinates

$$(1,-2,0),\, (1,-1,0) \,\,\,\,\mathrm{and}\,\,\,\,(0,1,1)\,.$$

Determine the three basis vectors $P_1(x),\,P_2(x)$ and $P_3(x)\,$.

Exercise 5: Subspaces. By Brain and Hand

C

Consider the set $\,G3\,$ of geometric vectors in space. Do subspaces in $\,G3\,$ with the dimensions 0, 1, 2, 3 or 4 exist? If they do exist describe them in words.

D

Is the set ${\,a\cos(x)+b\sin(x)\,|\,a,b\in\mathbb R\,}$ a subspace in $C^{0}(\mathbb R)\,$?

E

Is ${\,\left (x_1,x_2, x_3, x_4 \right)\, |\, x_1 \cdot x_2 \cdot x_3\cdot x_4=0 \,}$ a subspace in $\mathbb R^4\,?$

F

Is the subset of polynomials $\,P_2(\Bbb R)\,$ having the root 1, a subspace in $\,P_2(\Bbb R)\,?$ If so, find a basis for the subspace.

G

Is the subspace of polynomials in $\,P_2(\Bbb R)\,$ having a double root, at subspace in $\,P_2(\Bbb R)\,?$ If so, find a basis for the subspace.

Exercise 6: Bases for Subspaces

A

By hand: Explain that the solution set for the system of linear equations

$$ \begin{aligned} x_2 +3x_3 - x_4+2x_5 &= 0\\\\ 2x_1+3x_2+x_3+3x_4 &= 0\\\\ x_1 + x_2 -x_3 + 2x_4-x_5 &= 0 \end{aligned}$$

is a subspace in $\mathbb R^5\,$, state the dimension of the subspace, and determine a basis for this subspace.

B

Show that the two vectors

$$\ma_1=(1,0,1,0,1,0)\,\,\,\mathrm{and}\,\,\, \ma_2=(0,1,1,1,1,-1)$$

span the same subsapce in $\mathbb R^6$ as the vectors

$$\mb_1=(4,-5,-1,-5,-1,5) \,\,\,\mathrm{and}\,\,\,\mb_2=(-3,2,-1,2,-1,-2)\,.$$

%####### begin:hint %See Theorem 11.30 i eNote 11.

%####### end:hint

Exercise 7: Vectors Within and Outside a Subspace

C

Introductory meditation about identity: A matrix is a vector is a vector is a matrix (freely after Gertrud Stein, 1913: A rose is a rose is a rose is a rose).

In the vector space $\,\reel^{3\times 3}\,$ four vectors are given:

$$ \begin{matr}{rrr} 1 &0 & 0 \\\\ 0 & -2 & 0 \\\\ 0& 0 & 3 \end{matr}\,,\,\,\, \begin{matr}{rrr} 0 &-3 & 0 \\\\ 0 & 2 & 0 \\\\ 0 & -1 & 0 \end{matr}\,,\,\,\, \begin{matr}{rrr} 0 &0 & 1 \\\\ 0 & -2 & 0 \\\\ 3 & 0 & 0 \end{matr}\,,\,\,\, \begin{matr}{rrr} 0 &0 & 0 \\\\ -1 & 2 & -3\\\\ 0 & 0 & 0 \end{matr}\,.$$
D

Show that the four vectors are linearly independent.

We consider the subspace in $\,U\subset \reel^{3\times 3}\,$ that is spanned by the four vectors.

E

Choose a basis for $\,U\,,$ and show that

$$ \begin{matr}{rrr} 2 &-3 & -2 \\\\ -3 & 8 & -9 \\\\ -6& -1 & 6 \end{matr} \in U \,.$$

Determine the coordinate vector for this vector with respect to the chosen basis for $\,U\,.$

F

Find a vector $\,\mathbf v \in \reel^{3\times 3}\,$ that fulfills $\,\mathbf v \notin U\,.$

Exercise 8: Basis for Spanning (Advanced)

In $\, P_2(\Bbb R)\, $ the following vectors are given

$$ P_1(x) = 1 - 3x +2x^2, \; P_2(x) = 1 + x + 4x^2,\; P_3(x) = 1 -7x\,.$$
G

Show that $\, \, \big (P_1(x), P_2(x)\big ) \, $ is a basis for $\,\mathrm{span}{P_1(x), P_2(x), P_3(x)}$.

G

Investigate whether the vectors

$$\,Q_1(x) = 1 + 5x + 9x^2\,\,\,\,\mathrm{and}\,\,\,\, Q_2(x) = 3 - x +10x^2\,$$

belong to $\,\mathrm{span}{P_1(x), P_2(x), P_3(x)}\,$ and if so state the coordinate vectors with respect to the basis $\,\big (P_1(x), P_2(x)\big)\,$.

H

State the simplest possible basis for $\text{span}{P_1(x), P_2(x), P_3(x),Q_1(x),Q_2(x) }$.