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Section 3.3 Image and Kernel (AT3)

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Subsection 3.3.1 Class Activities

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Activity 3.3.1.

Let T:R2→R3 be given by

T([xy])=[xy0]with standard matrix [100100]

Which of these subspaces of R2 describes the set of all vectors that transform into 0→?

  1. {[aa]|a∈R}

  2. {[00]}

  3. R2={[xy]|x,y∈R}

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Definition 3.3.2.

Let T:V→W be a linear transformation, and let z→ be the additive identity (the “zero vector”) of W. The kernel of T is an important subspace of V defined by

ker⁡T={v→∈V | T(v→)=z→}
Figure 27. The kernel of a linear transformation
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Activity 3.3.3.

Let T:R3→R2 be given by

T([xyz])=[xy]with standard matrix [100010]

Which of these subspaces of R3 describes ker⁡T, the set of all vectors that transform into 0→?

  1. {[00a]|a∈R}

  2. {[aa0]|a∈R}

  3. {[000]}

  4. R3={[xyz]|x,y,z∈R}

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Activity 3.3.4.

Let T:R3→R2 be the linear transformation given by the standard matrix

T([xyz])=[3x+4y−zx+2y+z]
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(a)

Set T([xyz])=[00] to find a linear system of equations whose solution set is the kernel.

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(b)

Use RREF(A) to solve this homogeneous system of equations and find a basis for the kernel of T.
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Activity 3.3.5.

Let T:R4→R3 be the linear transformation given by

T([xyzw])=[2x+4y+2z−4w−2x−4y+z+w3x+6y−z−4w].

Find a basis for the kernel of T.
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Activity 3.3.6.

Let T:R2→R3 be given by

T([xy])=[xy0]with standard matrix [100100]

Which of these subspaces of R3 describes the set of all vectors that are the result of using T to transform R2 vectors?

  1. {[00a]|a∈R}

  2. {[ab0]|a,b∈R}

  3. {[000]}

  4. R3={[xyz]|x,y,z∈R}

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Definition 3.3.7.

Let T:V→W be a linear transformation. The image of T is an important subspace of W defined by

ImT={w→∈W | there is some v→∈V with T(v→)=w→}

In the examples below, the left example's image is all of R2, but the right example's image is a planar subspace of R3.

Figure 28. The image of a linear transformation
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Activity 3.3.8.

Let T:R3→R2 be given by

T([xyz])=[xy]with standard matrix [100010]

Which of these subspaces of R2 describes ImT, the set of all vectors that are the result of using T to transform R3 vectors?

  1. {[aa]|a∈R}

  2. {[00]}

  3. R2={[xy]|x,y∈R}

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Activity 3.3.9.

Let T:R4→R3 be the linear transformation given by the standard matrix

A=[3471−1102213−1]=[T(e→1)T(e→2)T(e→3)T(e→4)].

Since for a vector v→=[x1x2x3x4], T(v→)=T(x1e→1+x2e→2+x3e→3+x4e→4), which of the following best describes the set of vectors

{[3−12],[411],[703],[12−1]}?

  1. The set of vectors spans ImT but is not linearly independent.

  2. The set of vectors is a linearly independent subset of ImT but does not span ImT.

  3. The set of vectors is linearly independent and spans ImT; that is, the set of vectors is a basis for ImT.

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Observation 3.3.10.

Let T:R4→R3 be the linear transformation given by the standard matrix

A=[3471−1102213−1].

Since the set {[3−12],[411],[703],[12−1]} spans ImT, we can obtain a basis for ImT by finding RREFA=[101−101110000] and only using the vectors corresponding to pivot columns:

{[3−12],[411]}
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Activity 3.3.12.

Let T:R3→R4 be the linear transformation given by the standard matrix

A=[1−322−60001−131].

Find a basis for the kernel and a basis for the image of T.
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Activity 3.3.13.

Let T:Rn→Rm be a linear transformation with standard matrix A. Which of the following is equal to the dimension of the kernel of T?

  1. The number of pivot columns

  2. The number of non-pivot columns

  3. The number of pivot rows

  4. The number of non-pivot rows

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Activity 3.3.14.

Let T:Rn→Rm be a linear transformation with standard matrix A. Which of the following is equal to the dimension of the image of T?

  1. The number of pivot columns

  2. The number of non-pivot columns

  3. The number of pivot rows

  4. The number of non-pivot rows

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Observation 3.3.15.

Combining these with the observation that the number of columns is the dimension of the domain of T, we have the rank-nullity theorem:

The dimension of the domain of T equals dim⁡(ker⁡T)+dim⁡(ImT).

The dimension of the image is called the rank of T (or A) and the dimension of the kernel is called the nullity.

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Activity 3.3.16.

Let T:R3→R4 be the linear transformation given by the standard matrix

A=[1−322−60001−131].

Verify that the rank-nullity theorem holds for T.
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Subsection 3.3.2 Videos

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Figure 29. Video: The kernel and image of a linear transformation
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Figure 30. Video: Finding a basis of the image of a linear transformation
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Figure 31. Video: Finding a basis of the kernel of a linear transformation
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Figure 32. Video: The rank-nullity theorem
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Subsection 3.3.3 Slideshow

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Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/AT3.slides.html.

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Exercises 3.3.4 Exercises

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Exercises available at https://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/AT3/.

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Subsection 3.3.5 Mathematical Writing Explorations

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Exploration 3.3.17.

Assume f:V→W is a linear map. Let {v1→,v2→,…,vn→} be a set of vectors in V, and set wi→=f(vi→).

  • If the set {w1→,w2→,…,wn→} is linearly independent, must the set {v1→,v2→,…,vn→} also be linearly independent?

  • If the set {v1→,v2→,…,vn→} is linearly independent, must the set {w1→,w2→,…,wn→} also be linearly independent?

  • If the set {w1→,w2→,…,wn→} spans W, must the set {v1→,v2→,…,vn→} also span V?

  • If the set {v1→,v2→,…,vn→} spans V, must the set {w1→,w2→,…,wn→} also span W?

  • In light of this, is the image of the basis of a vector space always a basis for the codomain?

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Exploration 3.3.18.

Prove the Rank-Nullity Theorem. Use the steps below to help you.

  • The theorem states that, given a linear map h:V→W, with V and W vector spaces, the rank of h, plus the nullity of h, equals the dimension of the domain V. Assume that the dimension of V is n.

  • For simplicity, denote the rank of h by R(h), and the nullity by N(h).

  • Recall that R(h) is the dimension of the range space of h. State the precise definition.

  • Recall that N(h) is the dimension of the null space of h. State the precise definition.

  • Begin with a basis for the null space, denoted BN={β1→,β2→,…,βk→}. Show how this can be extended to a basis BV for V, with BV={β1→,β2→,…,βk→,βk+1→,βk+2→,…,βn→}. In this portion, you should assume k≤n, and construct additional vectors which are not linear combinations of vectors in BN. Prove that you can always do this until you have n total linearly independent vectors.

  • Show that BR={h(βk+1→),h(βk+2→),…,h(βn→)} is a basis for the range space. Start by showing that it is linearly independent, and be sure you prove that each element of the range space can be written as a linear combination of BR.

  • Show that BR spans the range space.

  • State your conclusion.

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Subsection 3.3.6 Sample Problem and Solution

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Sample problem Example B.1.16.

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