Linear algebraLesson 2 of 2 in the course Determinants and linear maps

Linear Maps: Matrices, Subspaces, and Structural Properties

5 flashcards · 5 quiz questions · 5 match pairs · 4 fill-in sentences · English

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Summary

This lesson introduces linear maps and the two rules that define them. These rules explain why linear maps preserve the structure of vector spaces.

T(c1u1+⋯+cnun)=c1T(u1)+⋯+cnT(un).T(c_1u_1+\cdots+c_nu_n)=c_1T(u_1)+\cdots+c_nT(u_n).

A linear map is therefore also called an operation-preserving map or a vector-space homomorphism.

Setting the scalar equal to zero in the homogeneity rule gives T(0V)=0WT(0_V)=0_W. A linear map cannot shift the origin to a different point. It also maps a subspace to a subspace, although the dimension of the image may be smaller than that of the original subspace.

All flashcards

What is a linear map?
A function between vector spaces over the same field that preserves vector addition and scalar multiplication.
How is a linear map represented by a matrix in chosen bases?
Column jj contains the coordinates of the image of the jjth domain basis vector.
When do prescribed images on a dependent set define a consistent linear extension?
They must preserve every linear relation among the vectors in the set.
What does the rank–nullity theorem state for a finite-dimensional domain?
For T:V→WT:V\to W, dim⁡(ker⁡T)+dim⁡(im⁡T)=dim⁡(V)\dim(\ker T)+\dim(\operatorname{im}T)=\dim(V).
What is the cokernel of a linear map T:V→WT:V\to W?
The quotient space W/im⁡(T)W/\operatorname{im}(T).

Quiz questions

  1. A linear map T:R4→R3T:\mathbb{R}^4\to\mathbb{R}^3 has rank 22. What is its nullity?Answer: 22
  2. Old and new coordinates satisfy [u]=B[u′][u]=B[u'], and AA is an endomorphism's old-basis matrix. What is its new-basis matrix?Answer: B−1ABB^{-1}AB
  3. Linear maps f:V→Wf:V\to W and g:W→Zg:W\to Z have matrices FF and GG in compatible bases. What matrix represents g∘fg\circ f?Answer: GFGF
  4. Which condition is equivalent to injectivity of a linear map T:V→WT:V\to W?Answer: ker⁡T={0}\ker T=\{0\}
  5. For a linear map T:V→WT:V\to W, which quotient space is its cokernel?Answer: W/im⁡(T)W/\operatorname{im}(T)

Based on the Wikipedia article Linear map, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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