TBIL-LA Subspace Basis and Dimension (VS7)

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Section 2.7 Subspace Basis and Dimension (VS7)

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Learning Outcomes

Compute a basis for the subspace spanned by a given set of Euclidean vectors, and determine the dimension of the subspace.

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Subsection 2.7.1 Class Activities

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Observation 2.7.1.

Recall from section Section 2.4 that a subspace of a vector space is a subset that is itself a vector space.

One easy way to construct a subspace is to take the span of set, but a linearly dependent set contains “redundant” vectors. For example, only two of the three vectors in the following image are needed to span the planar subspace.

Figure 19. A linearly dependent set of three vectors

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Activity 2.7.2.

Consider the subspace of $R^{4}$ given by $W = span {[\begin{matrix} 2 \\ 3 \\ 0 \\ 1 \end{matrix}], [\begin{matrix} 2 \\ 0 \\ 1 \\ - 1 \end{matrix}], [\begin{matrix} 2 \\ - 3 \\ 2 \\ - 3 \end{matrix}], [\begin{matrix} 1 \\ 5 \\ - 1 \\ 0 \end{matrix}]} .$

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(a)

Mark the part of $RREF [\begin{array}{cccc} 2 & 2 & 2 & 1 \\ 3 & 0 & - 3 & 5 \\ 0 & 1 & 2 & - 1 \\ 1 & - 1 & - 3 & 0 \end{array}]$ that shows that $W$ 's spanning set is linearly dependent.

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(b)

Find a basis for $W$ by removing a vector from its spanning set to make it linearly independent.


    
        
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rref([2,2,2,1; 3,0,-3,5; 0,1,2,-1; 1,-1,-3,0])

    
    
    
    
        
            
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Fact 2.7.3.

Let $S = {{\vec{v}}_{1}, \dots, {\vec{v}}_{m}} .$ The easiest basis describing $span S$ is the set of vectors in $S$ given by the pivot columns of $RREF [{\vec{v}}_{1} \dots {\vec{v}}_{m}] .$

Put another way, to compute a basis for the subspace $span S,$ simply remove the vectors corresponding to the non-pivot columns of $RREF [{\vec{v}}_{1} \dots {\vec{v}}_{m}] .$ For example, since

RREF [\begin{array}{ccc} 1 & 2 & 3 \\ 0 & - 2 & - 2 \\ - 3 & 1 & - 2 \end{array}] = [\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{array}]

the subspace $W = span {[\begin{matrix} 1 \\ 0 \\ - 3 \end{matrix}], [\begin{matrix} 2 \\ - 2 \\ 1 \end{matrix}], [\begin{matrix} 3 \\ - 2 \\ - 2 \end{matrix}]}$ has ${[\begin{matrix} 1 \\ 0 \\ - 3 \end{matrix}], [\begin{matrix} 2 \\ - 2 \\ 1 \end{matrix}]}$ as a basis.

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Activity 2.7.4.

Let $W$ be the subspace of $R^{4}$ given by

W = span {[\begin{matrix} 1 \\ 3 \\ 1 \\ - 1 \end{matrix}], [\begin{matrix} 2 \\ - 1 \\ 1 \\ 2 \end{matrix}], [\begin{matrix} 4 \\ 5 \\ 3 \\ 0 \end{matrix}], [\begin{matrix} 3 \\ 2 \\ 2 \\ 1 \end{matrix}]} .

Find a basis for $W .$

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Activity 2.7.5.

Let $W$ be the subspace of $P_{3}$ given by

W = span {x^{3} + 3 x^{2} + x - 1, 2 x^{3} - x^{2} + x + 2, 4 x^{3} + 5 x^{2} + 3 x, 3 x^{3} + 2 x^{2} + 2 x + 1}

Find a basis for $W .$

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Activity 2.7.6.

Let $W$ be the subspace of $M_{2, 2}$ given by

W = span {[\begin{array}{cc} 1 & 3 \\ 1 & - 1 \end{array}], [\begin{array}{cc} 2 & - 1 \\ 1 & 2 \end{array}], [\begin{array}{cc} 4 & 5 \\ 3 & 0 \end{array}], [\begin{array}{cc} 3 & 2 \\ 2 & 1 \end{array}]} .

Find a basis for $W .$

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Activity 2.7.7.

Let

S = {[\begin{matrix} 2 \\ 3 \\ 0 \\ 1 \end{matrix}], [\begin{matrix} 2 \\ 0 \\ 1 \\ - 1 \end{matrix}], [\begin{matrix} 2 \\ - 3 \\ 2 \\ - 3 \end{matrix}], [\begin{matrix} 1 \\ 5 \\ - 1 \\ 0 \end{matrix}]}

and

T = {[\begin{matrix} 2 \\ 0 \\ 1 \\ - 1 \end{matrix}], [\begin{matrix} 2 \\ - 3 \\ 2 \\ - 3 \end{matrix}], [\begin{matrix} 1 \\ 5 \\ - 1 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 3 \\ 0 \\ 1 \end{matrix}]} .

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(a)

Find a basis for $span S .$

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(b)

Find a basis for $span T .$

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Observation 2.7.8.

Even though we found different bases for them, $span S$ and $span T$ are exactly the same subspace of $R^{4},$ since

S = {[\begin{matrix} 2 \\ 3 \\ 0 \\ 1 \end{matrix}], [\begin{matrix} 2 \\ 0 \\ 1 \\ - 1 \end{matrix}], [\begin{matrix} 2 \\ - 3 \\ 2 \\ - 3 \end{matrix}], [\begin{matrix} 1 \\ 5 \\ - 1 \\ 0 \end{matrix}]} = {[\begin{matrix} 2 \\ 0 \\ 1 \\ - 1 \end{matrix}], [\begin{matrix} 2 \\ - 3 \\ 2 \\ - 3 \end{matrix}], [\begin{matrix} 1 \\ 5 \\ - 1 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 3 \\ 0 \\ 1 \end{matrix}]} = T .

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Fact 2.7.9.

Any non-trivial real vector space has infinitely-many different bases, but all the bases for a given vector space are exactly the same size.

For example,

{{\vec{e}}_{1}, {\vec{e}}_{2}, {\vec{e}}_{3}} and {[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]} and {[\begin{matrix} 1 \\ 0 \\ - 3 \end{matrix}], [\begin{matrix} 2 \\ - 2 \\ 1 \end{matrix}], [\begin{matrix} 3 \\ - 2 \\ 5 \end{matrix}]}

are all valid bases for $R^{3},$ and they all contain three vectors.

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Definition 2.7.10.

The dimension of a vector space is equal to the size of any basis for the vector space.

As you'd expect, $R^{n}$ has dimension $n .$ For example, $R^{3}$ has dimension $3$ because any basis for $R^{3}$ such as

{{\vec{e}}_{1}, {\vec{e}}_{2}, {\vec{e}}_{3}} and {[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]} and {[\begin{matrix} 1 \\ 0 \\ - 3 \end{matrix}], [\begin{matrix} 2 \\ - 2 \\ 1 \end{matrix}], [\begin{matrix} 3 \\ - 2 \\ 5 \end{matrix}]}

contains exactly three vectors.

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Activity 2.7.11.

Find the dimension of each subspace of $R^{4}$ by finding $RREF$ for each corresponding matrix.

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(a)

span {[\begin{matrix} 2 \\ 3 \\ 0 \\ - 1 \end{matrix}], [\begin{matrix} 2 \\ 0 \\ 0 \\ 3 \end{matrix}], [\begin{matrix} 4 \\ 3 \\ 0 \\ 2 \end{matrix}], [\begin{matrix} - 3 \\ 0 \\ 1 \\ 3 \end{matrix}]}

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(b)

span {[\begin{matrix} 2 \\ 3 \\ 0 \\ - 1 \end{matrix}], [\begin{matrix} 2 \\ 0 \\ 0 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 13 \\ 7 \\ 16 \end{matrix}], [\begin{matrix} - 1 \\ 10 \\ 7 \\ 14 \end{matrix}], [\begin{matrix} 4 \\ 3 \\ 0 \\ 2 \end{matrix}]}

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(c)

span {[\begin{matrix} 2 \\ 3 \\ 0 \\ - 1 \end{matrix}], [\begin{matrix} 4 \\ 3 \\ 0 \\ 2 \end{matrix}], [\begin{matrix} - 3 \\ 0 \\ 1 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 6 \\ 1 \\ 5 \end{matrix}]}

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(d)

span {[\begin{matrix} 5 \\ 3 \\ 0 \\ - 1 \end{matrix}], [\begin{matrix} - 2 \\ 1 \\ 0 \\ 3 \end{matrix}], [\begin{matrix} 4 \\ 5 \\ 1 \\ 3 \end{matrix}]}

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Subsection 2.7.2 Videos

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Figure 20. Video: Finding a basis of a subspace and computing the dimension of a subspace

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Exploration 2.7.12.

Prove each of the following statements is true.

If ${{\vec{b}}_{1}, {\vec{b}}_{2}, \dots, {\vec{b}}_{m}}$ and ${{\vec{c}}_{1}, {\vec{c}}_{2}, \dots, {\vec{c}}_{n}}$ are each a basis for a vector space $V,$ then $m = n .$
If ${{\vec{v}}_{1}, {\vec{v}}_{2} \dots, {\vec{v}}_{n}}$ is linearly independent, then so is ${{\vec{v}}_{1}, {\vec{v}}_{1} + {\vec{v}}_{2}, \dots, {\vec{v}}_{1} + {\vec{v}}_{2} + \dots + {\vec{v}}_{n}} .$
Let $V$ be a vector space of dimension $n,$ and $\vec{v} \in V .$ Then there exists a basis for $V$ which contains $\vec{v} .$

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Exploration 2.7.13.

Suppose we have the set of all function

f : S \to R .

We claim that this is a vector space under the usual operation of function addition and scalar multiplication. What is the dimension of this space for each choice of

S

below:

$S = {1}$
$S = {1, 2}$
$S = {1, 2, \dots, n}$
$S = R$

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Exploration 2.7.14.

Suppose you have the vector space

V = {(\begin{matrix} x \\ y \\ z \end{matrix}) \in R^{3} : x + y + z = 1}

with the operations

(\begin{matrix} x_{1} \\ y_{1} \\ z_{1} \end{matrix}) \oplus (\begin{matrix} x_{2} \\ y_{2} \\ z_{2} \end{matrix}) = (\begin{matrix} x_{1} + x_{2} - 1 \\ y_{1} + y_{2} \\ z_{1} + z_{2} \end{matrix}) and α ⊙ (\begin{matrix} x_{1} \\ y_{1} \\ z_{1} \end{matrix}) = (\begin{matrix} α x_{1} - α + 1 \\ α y_{1} \\ α z_{1} \end{matrix}) .

Find a basis for

V

and determine it's dimension.

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Subsection 2.7.6 Sample Problem and Solution

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Sample problem Example B.1.11.

You have attempted 1 of 1 activities on this page.

Section 2.7 Subspace Basis and Dimension (VS7)

Learning Outcomes

Subsection 2.7.1 Class Activities

Observation 2.7.1.

Activity 2.7.2.

(a)

(b)

Fact 2.7.3.

Activity 2.7.4.

Activity 2.7.5.

Activity 2.7.6.

Activity 2.7.7.

(a)

(b)

Observation 2.7.8.

Fact 2.7.9.

Definition 2.7.10.

Activity 2.7.11.

(a)

(b)

(c)

(d)

Subsection 2.7.2 Videos

Subsection 2.7.3 Slideshow

Exercises 2.7.4 Exercises

Subsection 2.7.5 Mathematical Writing Explorations

Exploration 2.7.12.

Exploration 2.7.13.

Exploration 2.7.14.

Subsection 2.7.6 Sample Problem and Solution