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Section 3.2 Standard Matrices (AT2)

Subsection 3.2.1 Class Activities

Remark 3.2.1.

Recall that a linear map T:VW satisfies

  1. T(v+w)=T(v)+T(w) for any v,wV.

  2. T(cv)=cT(v) for any cR,vV.

In other words, a map is linear when vector space operations can be applied before or after the transformation without affecting the result.

Activity 3.2.2.

Suppose T:R3R2 is a linear map, and you know T([100])=[21] and T([001])=[32]. What is T([300])?

  1. [63]

  2. [96]

  3. [42]

  4. [64]

Activity 3.2.3.

Suppose T:R3R2 is a linear map, and you know T([100])=[21] and T([001])=[32]. What is T([101])?

  1. [21]

  2. [31]

  3. [13]

  4. [58]

Activity 3.2.4.

Suppose T:R3R2 is a linear map, and you know T([100])=[21] and T([001])=[32]. What is T([203])?

  1. [21]

  2. [31]

  3. [13]

  4. [58]

Activity 3.2.5.

Suppose T:R3R2 is a linear map, and you know T([100])=[21] and T([001])=[32]. What piece of information would help you compute T([041])?

  1. The value of T([040]).

  2. The value of T([010]).

  3. The value of T([111]).

  4. Any of the above.

Definition 3.2.7.

Since a linear transformation T:RnRm is determined by its action on the standard basis {e1,,en}, it is convenient to store this information in an m×n matrix, called the standard matrix of T, given by [T(e1)T(en)].

For example, let T:R3R2 be the linear map determined by the following values for T applied to the standard basis of R3.

T(e1)=T([100])=[32]T(e2)=T([010])=[14]T(e3)=T([001])=[50]

Then the standard matrix corresponding to T is

[T(e1)T(e2)T(e3)]=[315240].

Activity 3.2.8.

Let T:R4R3 be the linear transformation given by

T(e1)=[032]T(e2)=[301]T(e3)=[421]T(e4)=[200]

Write the standard matrix [T(e1)T(en)] for T.

Activity 3.2.9.

Let T:R3R2 be the linear transformation given by

T([xyz])=[x+3z2xy4z]
(a)

Compute T(e1), T(e2), and T(e3).

(b)

Find the standard matrix for T.

Activity 3.2.11.

Let T:R3R3 be the linear transformation given by the standard matrix

[321452021].
(a)

Compute T([123]).

(b)

Compute T([xyz]).

Activity 3.2.12.

Compute the following linear transformations of vectors given their standard matrices.

(a)
T1([12]) for the standard matrix A1=[43011130]
(b)
T2([1103]) for the standard matrix A2=[43011130]
(c)
T3([020]) for the standard matrix A3=[430013511300]

Subsection 3.2.2 Videos

Figure 26. Video: Using the standard matrix to compute the image of a vector

Subsection 3.2.3 Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/AT2.slides.html.

Exercises 3.2.4 Exercises

Exercises available at https://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/AT2/.

Subsection 3.2.5 Mathematical Writing Explorations

We can represent images in the plane R2 using vectors, and manipulate those images with linear transformations. We introduce some notation in these explorations that is needed for their completion, but is not essential to the rest of the text. These have a geometric flair to them, and can be understood by thinking of geometric transformations in terms of standard matrices.

Given two vectors v=[v1v2vn] and w=[w1w2wn], we define the dot product as

vw=v1w1+v2w2+vnwn.

Exploration 3.2.13.

For each of the following properties, determine if it is held by the dot product. Either provide a proof it the property holds, or provide a counter-example if it does not.
  • Distributive over addition (e.g., (u+v)w=uw+vw)?

  • Associative?

  • Commutative?

Exploration 3.2.14.

Given the properties you proved in the last exploration, could the dot product take the place of as a vector space operation on Rn?

Exploration 3.2.15.

Is the dot product a linear operator? That is, given vectors u,v,wRn, and k,mR, is it true that
u(kv+mw)=k(uv)+m(uw).
Prove or provide a counter-example.

Exploration 3.2.16.

Assume v=[v1v2vn] and define the length of a vector by

|v|=(i=1nvi2)1/2.

Prove that |u|=|v| if an only if u+v and uv are perpendicular. You may use the fact (try and prove it!) that two vectors are perpendicular if and only if their dot product is zero.

Exploration 3.2.17.

  • A dilation is given by by mapping a vector v=[xy] to some scalar multiple of v.

  • A rotation is given by v[cos(θ)xsin(θ)ycos(θ)y+sin(θ)x].

  • A reflection of v over a line l can be found by first finding a vector l=[lxly] along l, then v2lvlllv.

Represent each of the following transformations with respect to the standard basis in R2.
  • Rotation through an angle θ.

  • Reflection over a line l passing through the origin.

  • Dilation by some scalar s.

Prove that each transformation is linear, and that your matrix representations are correct.

Subsection 3.2.6 Sample Problem and Solution

Sample problem Example B.1.15.

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