Section 4.1 Subspaces and Spanning

Definition: A subspace of ${\mathbb{R}}^n$ is any set H in ${\mathbb{R}}^n$ that has three properties:

a) The zero vector is in H.

b) For each $\overrightarrow{u}$ and $\overrightarrow{v}$ in H, the sum $\overrightarrow{u}+\overrightarrow{v}$ is in H.

c) For each $\overrightarrow{u}$ in H and each scalar $c$ in $\mathbb{R}$, the vector $c\overrightarrow{u}$ is in H.

Fact: The subspace definition is geometric, a plane through the origin is a subspace of ${\mathbb{R}}^3$.

Example 1: Let $H=\left\{\begin{array}{c}\left[\begin{array}{c}x\\ y\end{array}\right]:x\ge 0,y\ge 0\end{array}\right\}$ the first quadrant in the $xy$-plane. Show (a): if $\overrightarrow{u}$ and $\overrightarrow{v}$ are in H then $\overrightarrow{u}+\overrightarrow{v}$ are in H, (b): Find a vector $\overrightarrow{u}$ and a scalar $c$ such that $c\overrightarrow{u}$ is not in H ( this shows V is not a subspace of ${\mathbb{R}}^2$).

 

 

Exercise 1: Let $H=\left\{\begin{array}{c}\left[\begin{array}{c}x\\ y\end{array}\right]:xy\ge 0,\end{array}\right\}$ the union of first quadrant and the third quadrant in the $xy$-plane. Show (a): if $\overrightarrow{u}$ in H and any scalar $c$, then $c\overrightarrow{u}$ is in H, (b): Find vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ in H such that $\overrightarrow{u}+\overrightarrow{v}$ is not in H.

 

Fact: 1. The set only has zero vector, $\overrightarrow{0}$ is a subspace of ${\mathbb{R}}^n$ and we call it zero subspace which is written as $\left\{\begin{array}{c}\overrightarrow{0}\end{array}\right\}$.

2. ${\mathbb{R}}^2$ is NOT a subspace of ${\mathbb{R}}^3$ but $H=\left\{\begin{array}{ccc}\left[\begin{array}{c}a\\ b\\ 0\end{array}\right]:a,b& \text{in}& {\mathbb{R}}^n\end{array}\right\}$ is a subspace of ${\mathbb{R}}^3$.

 

Theorem: If $\overrightarrow{x_1},\cdots ,\overrightarrow{x_p}$ are vectors in ${\mathbb{R}}^n$, then Span$\left\{\begin{array}{c}\overrightarrow{x_1},\cdots ,\overrightarrow{x_p}\end{array}\right\}$ is a subspace of ${\mathbb{R}}^n$.

Proof:

 

 

Example 2: Let H be the set of all vectors of the form $(2a+3b,-a+2b,2a,-b)$ in ${\mathbb{R}}^4$ where $a,b$ are in $\mathbb{R}$. Show H is a subspace of ${\mathbb{R}}^4$.

 

 

Exercise 2: Let H be the set of all vectors of the form $(s+4t,-s+t,2s+t,s-2t)$ in ${\mathbb{R}}^4$ where $s,t$ are in $\mathbb{R}$. Show H is a subspace of ${\mathbb{R}}^4$.

 

Example 3: Let W be the set of all vectors of the form shown where $a,b,c$ are scalars. Either find a set of vectors that spans W or show W is not a subspace.

(a) $\left[\begin{array}{c}2a+b\\ -3a+b\\ 2\end{array}\right]$ (b) $\left[\begin{array}{c}-a+b-c\\ 2a+b\\ 4c\\ 3b+c\end{array}\right]$.

 

 

Exercise 3: Let W be the set of all vectors of the form shown where $a,b,c$ are scalars. Either find a set of vectors that spans W or show W is not a subspace.

(a) $\left[\begin{array}{c}1\\ -a+b\\ 2a-b\end{array}\right]$ (b) $\left[\begin{array}{c}-a+2b-c\\ a-b\\ 0\\ 3b+3c\end{array}\right]$.

 

Definition: The column space of an $m\times n$ matrix $A$ is the set Col $A$ of all linear combinations of the columns of $A$. The image space of an $m\times n$ matrix $A$, denote Im$A$, are defined by Im$A=\left\{\begin{array}{ccc}A\overrightarrow{x}|\overrightarrow{x}& \text{in}& {\mathbb{R}}^n\end{array}\right\}$. Im$A=$ Col $A$ is a subspace of ${\mathbb{R}}^n$.

 

Fact: 1. If $A=\left[\begin{array}{c}\overrightarrow{a_1}\cdots \overrightarrow{a_n}\end{array}\right]$ with the columns in ${\mathbb{R}}^m$ then Col$A$ is the same as Span$\left\{\begin{array}{c}\overrightarrow{a_1},\cdots ,\overrightarrow{a_n}\end{array}\right\}$ a subspace of ${\mathbb{R}}^m$.

2. The vector $\overrightarrow{b}$ is a linear combination of the columns of $A$ if and only if $\overrightarrow{b}$ can be written as $A\overrightarrow{x}$ for some $\overrightarrow{x}$, that is, if and only if the equation $A\overrightarrow{x}=\overrightarrow{b}$ has a solution if and only if $A\overrightarrow{x}=\overrightarrow{b}$ is consistent and $\overrightarrow{b}$ is in Col$A$.

 

 

Example 4: Show $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$ is in the Col$A$ where $A=\left[\begin{array}{ccc}1& 2& -1\\ 3& 0& 2\\ 0& -1& 2\end{array}\right]$.

 

 

Exercise 4: Show $\left[\begin{array}{c}2\\ 1\\ -1\end{array}\right]$ is in the Col$A$ where $A=\left[\begin{array}{ccc}2& 1& 1\\ 1& 0& -2\\ 0& -1& 2\end{array}\right]$.

 

Example 5: Find the values $h$ such that $\overrightarrow{y}$ is in the subspace span by $\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}$ where $\overrightarrow{y}=\left[\begin{array}{c}1\\ 2\\ h\end{array}\right],\overrightarrow{x_1}=\left[\begin{array}{c}-1\\ 0\\ 3\end{array}\right],\overrightarrow{x_2}=\left[\begin{array}{c}0\\ 2\\ 4\end{array}\right]$ and $\overrightarrow{x_3}=\left[\begin{array}{c}-2\\ 4\\ 14\end{array}\right]$.

 

 

Exercise 5: Find the values $h$ such that $\overrightarrow{y}$ is in the subspace span by $\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}$ where $\overrightarrow{y}=\left[\begin{array}{c}1\\ h\\ 3\end{array}\right],\overrightarrow{x_1}=\left[\begin{array}{c}0\\ 1\\ 3\end{array}\right],\overrightarrow{x_2}=\left[\begin{array}{c}3\\ 0\\ 4\end{array}\right]$ and $\overrightarrow{x_3}=\left[\begin{array}{c}6\\ -2\\ 2\end{array}\right]$.

 

Definition: The null space of a matrix $A$ is the set null $A$

(or Nul $A$) of all solutions of the homogenous equation $A\overrightarrow{x}=\overrightarrow{0}.$

 

Theorem: The null space of an $m\times n$ matrix $A$ is a subspace of ${\mathbb{R}}^n$. Equivalently, the set of all solutions of a system $A\overrightarrow{x}=\overrightarrow{0}$ of $m$ homogenous linear equations in $n$ unknowns is a subspace of ${\mathbb{R}}^n$.

 

Example6: Find the null space of $A=\left[\begin{array}{ccc}1& -1& 2\\ -4& 4& 5\\ -2& 2& 9\end{array}\right]$.

 

 

Exercise6: Find the null space of $A=\left[\begin{array}{ccc}3& -1& 1\\ 9& -7& -8\\ 2& -2& -3\end{array}\right]$.

 

Remark: $E_{\lambda }(A)=\text{null}(A-\lambda I)$ is a subspace of ${\mathbb{R}}^n$ for each $n\times n$ matrix $A$ and number $\lambda$. $\lambda$ is an eigenvalue of $A$ if $E_{\lambda }(A)\ne \{\overrightarrow{0}\}$ and $E_{\lambda }(A)$ called the eigenspace of $A$ corresponding $\lambda$ .The nonzero vectors in $E_{\lambda }(A)$  are the eigenvectors of $A$ corresponding to $\lambda$.

 

Example7: Find $E_2(A)=\text{null}(A-2I)$ where $A=\left[\begin{array}{ccc}A=2& 0& 0\\ 1& 2& -1\\ 1& 3& -2\end{array}\right]$.

 

 

Exercise7: Find $E_{-1}(A)=\text{null}(A+I)$ where $A=\left[\begin{array}{ccc}A=0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right]$.

 

 

GroupWork 1: True or False. Justify each answer:

a. A subspace of ${\mathbb{R}}^n$ is any set $H$ such that (i) the zero vector is in $H$, (ii) $\overrightarrow{u},\overrightarrow{v}$ and $\overrightarrow{u}+\overrightarrow{v}$ are in $H$ (iii) $c$ is a scalar, $c\overrightarrow{u}$ is in $H$.

 

b. If $\overrightarrow{v_1},\cdots ,\overrightarrow{v_p}$ are in ${\mathbb{R}}^n$, then span$\left\{\begin{array}{c}\overrightarrow{v_1},\cdots ,\overrightarrow{v_p}\end{array}\right\}$ is the same as the column space of the matrix $\left[\begin{array}{c}\overrightarrow{v_1},\cdots ,\overrightarrow{v_p}\end{array}\right]$.

 

c. The set of all solutions of a system of $m$ homogeneous equations in $n$ unknowns is a subspace of ${\mathbb{R}}^m$.

 

d. The null space of an $m\times n$ matrix is a subspace of ${\mathbb{R}}^n$.

 

e. The column space of a matrix $A$ is the set of solutions of $A\overrightarrow{x}=\overrightarrow{b}$.

 

GroupWork 2: Construct a nonzero $3\times 3$ matrix $A$ and a nonzero vector $\overrightarrow{b}$ such that $\overrightarrow{b}$ is in Col$A$, but $\overrightarrow{b}$ is not the same as any one of the columns of $A$.

 

GroupWork 3: Suppose $F$ is a $5\times 5$ matrix whose column space is not equal to ${\mathbb{R}}^5$. What can you say about Nul$F$?

 

GroupWork 4: In each case either show that the statement is true or give an
example showing that it is false.

a. If $U\ne {\mathbb{R}}^n$ is a subspace of ${\mathbb{R}}^n$ and $\overrightarrow{u}+\overrightarrow{v}$ is in $U$ then $\overrightarrow{u}$ and $\overrightarrow{v}$ are both in $U$.

 

b. If $U$ is a subspace of ${\mathbb{R}}^n$ and $r\overrightarrow{u}$ is in $U$ for all $r$ in $\mathbb{R}$ then $\overrightarrow{u}$ is in $U$.

 

c. If $U$ is a subspace of ${\mathbb{R}}^n$ and $\overrightarrow{u}$ is in $U$ then $-\overrightarrow{u}$ is also in $U$.

 

d. If $\overrightarrow{x}$ is in $U$ and $U=$ span$\left\{\begin{array}{c}\overrightarrow{y},\overrightarrow{z}\end{array}\right\}$, then $U=$ span$\left\{\begin{array}{c}\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}\end{array}\right\}$.

 

e. The empty set of vectors in ${\mathbb{R}}^n$ is a subspace of ${\mathbb{R}}^n$.

 

f. $\left[\begin{array}{c}0\\ 1\end{array}\right]$ is in span$\left\{\begin{array}{c}\left[\begin{array}{c}1\\ 0\end{array}\right],\left[\begin{array}{c}2\\ 0\end{array}\right]\end{array}\right\}$.

 

GroupWork 5: Construct a nonzero $3\times 3$ matrix $A$ and a vector $\overrightarrow{b}$ such that $\overrightarrow{b}$ is not in Col$A$.

 

GroupWork 6: If $A$ is a $5\times 5$ matrix and Nul$A$ is the zero subspace, what can you say about solutions of equations of the form $A\overrightarrow{x}=\overrightarrow{b}$ for $\overrightarrow{b}$ in ${\mathbb{R}}^5$?