how-many-rows-and-columns-must-a-matrix-a-have-in-order-to-define-a-mapping-from-mathbbr4-into-mathb

Name: how many rows and columns must a matrix a have in order to define a mapping from mathbbr4 into mathb
Uploaded: 2019-08-07
Duration: 1 min 39 s
Channel: Numerade
Description: How many rows and columns must a matrix $A$ have in order to define a mapping from $\mathbb{R}^{4}$ into $\mathbb{R}^{5}$ by the rule $T(\mathbf{x})=A \mathbf{x} ?$

Question

How many rows and columns must a matrix $A$ have in order to define a mapping from $\mathbb{R}^{4}$ into $\mathbb{R}^{5}$ by the rule $T(\mathbf{x})=A \mathbf{x} ?$

Ashley Boni · Accepted Answer

So we&#x27;re dealing with a function defined by the cruelty of X equals a X, and we want this function to be a mapping. Ah, from our four and two are five. So we want to start off with the metric, um, a vector of length for and end up with the vector of length five. So we needed to determine the size of a So say, this is our matrix A. We need to determine how many rows and how many columns we need. So the number of rose is determined by the CO domain, which is the range. And so the order of the range has five. Because this is our range and the number of columns is determined by the order of the domain, which is we&#x27;re starting off with, Ah, a domain of order for So that tells me that we need to have five rows and four columns. So a is going to be a, uh, five by four matrix

Viswesh Uppalapati · Answer

in this video, we&#x27;re gonna be selling question number 28 in section 1.7, which is based on then your independence. Even though this question is part of 1.7, this is ah, actual span question and it&#x27;s convey Usually answer using Dirham for which we learned in a previous chapter. So, uh, hearing for as in part of the span chapter, I don&#x27;t remember what section, but, um, basically what it states is that for? For matrix for an end by M matrix to span or n by n matrix to span um, r m so here in my anus, five by seven. So our m is, um ah, five is the set of all real numbers. Um, there is not a set of our room numbers, but the fifth dimension, as the column of the each column of every vector has each column address of five entries. So here a column vector of a five by seven would have 12345 entries because there are five rows. So basically what three and four says is that us This matrix, the five by seven matrix would span are five. If there was a pivot if there was a leading pivot in every row. So that means that there must be five pivot positions. Um, in each of the rose, you are in every room, so there should be a pivot here. Um, and seven non zero are zeros. So we have 1234567 Um, says you&#x27;re, uh, 4001 Is there, uh, 001 You&#x27;re, uh, so these are all the positions while the rest of them can be stars or zeroes. Okay. Um, so here we have five columns that have leading prohibitions. Uh, this is because this because for a set of matrices, suspend our, uh, here for a five by seven matrix. Suspend our five. We need a pivot position in every road. That means that we&#x27;ll have five pivot columns. So pivot column is just ah, column that holds a position. And, um

Henry Carnick · Answer

but we need to determine the nature trip visitation off our transformation. But the person we need to do is choose a basis for our domain on our coda mein. Since they&#x27;re both are two, we&#x27;re going to the simplest basis, which is the standard basis, which is 10 and 01 We now need to determine the image of the elements of our base other basis under the transformation. It&#x27;s in this case t of 10 is going to give us three times one minus two times zero comma, one plus five times zero, which is equal to 31 And for his 01 we get is equal to three times zero minus two times one comma zero plus five times one which is equal to their to five. Now, we need to write the images of these trends of these basic of these basis elements as corner vectors in terms of the base under the basis off our coda name. In this case, we&#x27;re gonna be writing them in terms of the standard basis, so 31 is gonna equal 30 plus zero one, which is equal three times 10 plus one times +01 and *** to five is equal to negative to zero plus zero five, which is equal to native to 10 Close 501 So now our corner vectors are going to be 31 and negative 25 And these corner vectors are going are going to be our column. Vectors for our matrix or matrix is going to be 31 Yeah, to five. So this is the Matrix representation off our linear transformation.

Henry Carnick · Answer

so to solve this problem we first knew choose a basis for the domain of transformation. Since the domain of our transformation is our three, we&#x27;re going to choose a standard basis for three. But is it gonna be 100 comma 010 comma zero 01 Next, we need to compute the image of each of these elements of the basis. Under our transformation, she&#x27;s gonna go something like this. T of 100 is equal to one plus five times zero minus three times zero is equal. One Teoh 010 is gonna equal zero plus five times one minus three times there, which is equal to five and t off. 00 one is gonna equal euro plus five times zero minor three times one, We&#x27;re sequel minus three. So now we can compute the mixture representation of our linear transformation, which is going to be one by minus three

Nez Nikoo · Answer

Okay. The problem is giving us a three by three matrix, and it&#x27;s asking for a trace of a I&#x27;ve already written here. Um, the question, because it makes it easier to assure you to trace of a basically is thes diagonal elements into would read so you can see it. All you have to do is at those which is to plus two plus negative five. And that&#x27;s very easy to plus trees for minus five. It&#x27;s the answer is gonna be minus one. So that is the trace A for this problem. Thank you.

Subham Jyoti Mishra · Answer

So in the given question, you have Emma tricks live minus seven three. Andi toe minus four. It is a mattress we have in the question now in the person. I&#x27;m sorry. This is minus one. Dave asked. Find out. Harming needles are there in the mattress. And how many court? Uncertain. So the number of rows is one. And the stool and about problems is run boots on three. So number off Whose is Do and problems? Three. So this is the answer given question. Do those and three bottoms.

Problem

Find all $\mathbf{x}$ in $\mathbb{R}^{4}$ that ar…

Problem 8 Easy Difficulty

How many rows and columns must a matrix $A$ have in order to define a mapping from $\mathbb{R}^{4}$ into $\mathbb{R}^{5}$ by the rule $T(\mathbf{x})=A \mathbf{x} ?$

Answer

5 Rows and 4 Columns.

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Catherine R.

Anna Marie V.

Caleb E.

Maria G.

Absolute Value - Example 1

Absolute Value - Example 2

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5 Rows and 4 Columns.

Linear Algebra and Its Applications

Catherine R.

Anna Marie V.

Caleb E.

Maria G.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Catherine R.

Anna Marie V.

Caleb E.

Maria G.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications