TBIL-LA Subspaces (VS4)

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Section 2.4 Subspaces (VS4)

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Learning Outcomes

Determine if a subset of $R^{n}$ is a subspace or not.

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Subsection 2.4.1 Class Activities

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Activity 2.4.1.

Consider two non-colinear vectors in $R^{3} .$ If we look at all linear combinations of those two vectors (that is, their span), we end up with a plane within $R^{3} .$ Call this plane $S .$

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(a)

Are all of the vectors in $S$ also in $R^{3} ?$

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(b)

Let $\vec{z}$ be the additive identity in $R^{3} .$ Is $\vec{z} \in S ?$

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(c)

For any unspecified $\vec{u}, \vec{v} \in S,$ is it the case that $\vec{u} + \vec{v} \in S ?$

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(d)

For any unspecified $\vec{u} \in S$ and $c \in R,$ is it the case that $c \vec{u} \in S ?$

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Definition 2.4.2.

A subset of a vector space is called a subspace if it is a vector space on its own. The operations of addition and scalar from the parent vector space are inherited by the subspace.

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Observation 2.4.3.

Note the similarities between a planar subspace spanned by two non-colinear vectors in $R^{3},$ and the Euclidean plane $R^{2} .$ While they are not the same thing (and shouldn't be referred to interchangably), algebraists call such similar spaces isomorphic; we'll learn what this means more carefully in a later chapter.

A planar subset of $\IR^3$ compared with the plane $\IR^2\text{.}$ — Figure 13. A planar subset of $R^{3}$ compared with the plane $R^{2} .$

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Fact 2.4.4.

Any subset $S$ of a vector space $V$ that contains the additive identity $\vec{0}$ satisfies the eight vector space properties in Definition 2.1.3 automatically, since the operations were well-defined for the parent vector space.

However, to verify that it's a subspace, we still need to check that addition and multiplication still make sense using when only vectors from $S$ are allowed to be used. So we need to check two things:

The set is closed under addition: for any $\vec{u}, \vec{v} \in S,$ the sum $\vec{u} + \vec{v}$ is also in $S .$
The set is closed under scalar multiplication: for any $\vec{u} \in S$ and scalar $c \in R,$ the product $c \vec{u}$ is also in $S .$

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Activity 2.4.5.

Let $S = {[\begin{matrix} x \\ y \\ z \end{matrix}] | x + 2 y + z = 0} .$

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(a)

Let $\vec{v} = [\begin{matrix} x \\ y \\ z \end{matrix}]$ and $\vec{w} = [\begin{matrix} a \\ b \\ c \end{matrix}]$ be vectors in $S,$ so $x + 2 y + z = 0$ and $a + 2 b + c = 0 .$ Show that $\vec{v} + \vec{w} = [\begin{matrix} x + a \\ y + b \\ z + c \end{matrix}]$ also belongs to $S$ by verifying that $(x + a) + 2 (y + b) + (z + c) = 0 .$

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(b)

Let $\vec{v} = [\begin{matrix} x \\ y \\ z \end{matrix}] \in S,$ so $x + 2 y + z = 0 .$ Show that $c \vec{v} = [\begin{matrix} c x \\ c y \\ c z \end{matrix}]$ also belongs to $S$ for any $c \in R$ by verifying an appropriate equation.

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(c)

Is $S$ is a subspace of $R^{3} ?$

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Activity 2.4.6.

Let $S = {[\begin{matrix} x \\ y \\ z \end{matrix}] | x + 2 y + z = 4} .$ Choose a vector $\vec{v} = [\begin{matrix} ? \\ ? \\ ? \end{matrix}]$ in $S$ and a real number $c = ?,$ and show that $c \vec{v}$ isn't in $S .$ Is $S$ a subspace of $R^{3} ?$

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Remark 2.4.7.

Since $0$ is a scalar and $0 \vec{v} = \vec{z}$ for any vector $\vec{v},$ a nonempty set that is closed under scalar multiplication must contain the zero vector $\vec{z}$ for that vector space.

Put another way, you can check any of the following to show that a nonempty subset $W$ isn't a subspace:

Show that $\vec{0} \notin W .$
Find $\vec{u}, \vec{v} \in W$ such that $\vec{u} + \vec{v} \notin W .$
Find $c \in R, \vec{v} \in W$ such that $c \vec{v} \notin W .$

If you cannot do any of these, then $W$ can be proven to be a subspace by doing the following:

Prove that $\vec{u} + \vec{v} \in W$ whenever $\vec{u}, \vec{v} \in W .$
Prove that $c \vec{v} \in W$ whenever $c \in R, \vec{v} \in W .$

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Activity 2.4.8.

Consider these subsets of $R^{3} :$

R = {[\begin{matrix} x \\ y \\ z \end{matrix}] | y = z + 1} S = {[\begin{matrix} x \\ y \\ z \end{matrix}] | y = | z |} T = {[\begin{matrix} x \\ y \\ z \end{matrix}] | z = x y} .

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(a)

Show $R$ isn't a subspace by showing that $\vec{0} \notin R .$

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(b)

Show $S$ isn't a subspace by finding two vectors $\vec{u}, \vec{v} \in S$ such that $\vec{u} + \vec{v} \notin S .$

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(c)

Show $T$ isn't a subspace by finding a vector $\vec{v} \in T$ such that $2 \vec{v} \notin T .$

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Activity 2.4.9.

Consider these subsets of $M_{2 \times 2},$ the vector space of all $2 \times 2$ matrices with real entries. Show that each of these sets is or is not a subspace of $M_{2 \times 2} .$

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(a)

{[\begin{array}{cc} a & 0 \\ 0 & b \end{array}] | a, b \in R} .

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(b)

{[\begin{array}{cc} a & 0 \\ 0 & b \end{array}] | a + b = 0} .

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(c)

{[\begin{array}{cc} a & 0 \\ 0 & b \end{array}] | a + b = 5} .

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(d)

{[\begin{array}{cc} a & c \\ 0 & b \end{array}] | a + b = 0, c \in R} .

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Activity 2.4.10.

Let $W$ be a subspace of a vector space $V .$ How are $span W$ and $W$ related?

$span W$ may include vectors that aren't in $W$
$W$ may include vectors that aren't in $span W$
$W$ and $span W$ always contain the same vectors

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Fact 2.4.11.

If $S$ is any subset of a vector space $V,$ then since $span S$ collects all possible linear combinations, $span S$ is automatically a subspace of $V .$

In fact, $span S$ is always the smallest subspace of $V$ that contains all the vectors in $S .$

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Subsection 2.4.2 Videos

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Figure 14. Video: Showing that a subset of a vector space is a subspace

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Figure 15. Video: Showing that a subset of a vector space is not a subspace

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Exploration 2.4.12.

A square matrix

M

is symmetric if, for each index

i, j,

the entries

m_{i j} = m_{j i} .

That is, the matrix is itself when reflected over the diagonal from upper left to lower right. Prove that the set of

n \times n

symmetric matrices is a subspace of

M_{n \times n} .

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Exploration 2.4.13.

The space of all real-valued function of one real variable is a vector space. First, define

\oplus

and

⊙

for this vector space. Check that you have closure (both kinds!) and show what the zero vector is under your chosen addition. Decide if each of the following is a subspace. If so, prove it. If not, provide the counterexample.

The set of even functions, ${f : R \to R : f (- x) = f (x) for all x} .$
The set of odd functions, ${f : R \to R : f (- x) = - f (x) for all x} .$

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Exploration 2.4.14.

Give an example of each of these, or explain why it's not possible that such a thing would exist.

A nonempty subset of $M_{2 \times 2}$ that is not a subspace.
A set of two vectors in $R^{2}$ that is not a spanning set.

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Exploration 2.4.15.

Let

V

be a vector space and

S = {{\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{n}}

a subset of

V .

Show that the span of

S

is a subspace. Is it possible that there is a subset of

V

containing fewer vectors than

S,

but whose span contains all of the vectors in the span of

S ?

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Subsection 2.4.6 Sample Problem and Solution

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Sample problem Example B.1.8.

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