True or False: Linear Transformation & Matrix

Question

Determine which are true and which are false:

A) If there is a path of lenght 2 between two specific vertices in an undirected graph, then there must be a path of lenght 3 between those two vertices.

B) The adjacency matrix of A of every undirected graph has the property of A = A^T

C) If A ∈ M_2,2,, B ∈ M_2,3, and C ∈ M_3,4 then the following vlock matrix multiplication makes sense:

_ _ _ _

| A I₃ | | O I₃ |

| I₄ O | | I₄ A^T |

_ _ _ _

D) If A ∈ M_2,2,, B ∈ M_2,3, and C ∈ M_3,4 then the following vlock matrix multiplication makes sense:

_ _ _ _

| A B | | I₃ B |

| O C^T | | B^T O |

_ _ _ _

E) The standard matrix of a linear transformation T : R³ -> R⁷ has size 3 x 7.

Answer 1

A: consider the graph U --x-- V --y-- W. Where U V and W are vertices, x and y are edges. Any path from U to W must go through both x and y. The path can be lengthened by going backwards, but any step backwards must be matched by a step forwards. Hence every path from U to W is of even length, and xy is a path of length 2.

B: the entry a_ij in an adjacency matrix is 1 if there is an edge from i to j and 0 if there isnt an edge. since the graph is undirected, an edge from i to j exists if and only if there is an edge from j to i. In other words, a_ij = a_ji. The transpose of a matrix swaps the ij-th entry with the ji-th entry but as we just saw these are equal. So the adjacency matrix equals its transpose

(i am going to skip to E first since it will help with the next questions)

E: Remember what a matrix represents: the 1st column describes where the 1st basis vector is sent, the 2nd column describes where the 2nd basis vector is sent and so on. If you want to map from a 3 dimensional space, you have describe where 3 basis vectors go, so thats 3 columns. Each of these basis vectors will get sent into a space of dimension 7, so you need 7 rows do describe the output. so the matrix will have size 7x3.

C: Matricies correspond to linear transformations, and multiplying them corresponds to composing those transformations. Lets say you have two transformations: S : Rⁿ --> R^q and T : R^p --> R^m . If you want to do S and then do T, you need p = q. The composite is written as TS, so we will have T be an mxp matrix and S a pxn matrix. The result will be an mxn matrix.

For block matrix multiplication, we need every sub-multiplication to match up in the same way. Immediately we have AO + I₃I₄ . I₃I₄ doesnt make any sense.

D. By the same reasoning this is false. the first multiply here is AI₃ , which is a 2x2 being multiplied by a 3x3