# Linear algebra identity matrix

• ME_student
In summary: For question C: An nxn matrix A can be thought of as a transformation from R^n to R^n. If this transformation has a solution for every b in R^n, then it is an onto transformation. This means that the columns of A span R^n which also implies that they are linearly independent.For question D: I think there may be a typo in the original question. It should say "If the equation Ax=0 has a nontrivial solution, then A has AT LEAST n pivot positions." This is because if A is a square matrix, then the number of pivot positions will always be equal to the number of columns (which is also equal to the number of rows).The original
ME_student

## Homework Statement

A. If the equation Ax=0 has only the trivial solution, then A is row equivalent to the nxn identity matrix.

B. If the columns of A span R^n, the columns are linearly independent.

C. If A is an nxn matrix, then the equation Ax=b has at least one solution for each b in R^n.

D. If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions.

E. If A^T is not invertible, then A isn't invertible?

## The Attempt at a Solution

A. True

B. False

D. True I think.

As for the rest of them I would like to discuss them with you.

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You have a 50/50 chance of getting the right answer simply by guessing. You need to explain your reasoning.

For E: Perhaps thinking about the contrapositive would be easier.
For C: If A is an nxn matrix with real coefficients, then A is the matrix corresponding to some transformation T from Rn to Rn. If Ax = b has a solution for all b $\in$ Rn, what can you say about the properties of T?

kduna said:
For E: Perhaps thinking about the contrapositive would be easier.
For C: If A is an nxn matrix with real coefficients, then A is the matrix corresponding to some transformation T from Rn to Rn. If Ax = b has a solution for all b $\in$ Rn, what can you say about the properties of T?

The contrapositive? I have never heard of that before in my class.

By the way, question A. is suppose to say identity matrix, not matrix.

A. True because two matrices will have the same solution set. Ax=0 is the trivial solution, when augmenting the identity matrix with zeros the solution set will be equal to that of Ax=b.

B.False because we could have three independent vectors in R^3, although only two of them span a plane in R^3 while the other vector depends on one of the two. This is the reason I said false.

C.True because an nxn matrix has rows=columns, thus there must be a solution for each b in R^n.

D. True because the non trivial solution has a linear dependence relation. For example, let say we have a 3x4 matrix row 3 would have all zeros meaning x_3=x_3 (free) which depends on one of the other two vectors. Therefore the only two pivot positions will be in row 1 and row 2.

E. True because A has to be an invertible matrix to make A^T to be invertible. (A^-1)^-1=A

Is A supposed to be a square matrix? It seems like the question assumes it's nxn, but you didn't mention anything about the dimensions of A.

vela said:
Is A supposed to be a square matrix? It seems like the question assumes it's nxn, but you didn't mention anything about the dimensions of A.

Yeah nxn is suppose to be a square matrix. In our textbook nxn means square matrix mxn means non-square matrix.

ME_student said:
B.False because we could have three independent vectors in R^3, although only two of them span a plane in R^3 while the other vector depends on one of the two.

Then they're not independent, are they?

Any two linearly independent vectors in $\mathbb{R}^3$ will span a plane, but to span the whole of $\mathbb{R}^3$ you need another vector which doesn't lie in that plane. That vector is then linearly independent of both the first two.

C.True because an nxn matrix has rows=columns, thus there must be a solution for each b in R^n.

It is not stated that the matrix must be invertible. Thus the following is a counterexample: there is no $\vec x \in \mathbb{R}^2$ such that
$$\begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix} \vec x = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.$$

ME_student said:
The contrapositive? I have never heard of that before in my class.

By the way, question A. is suppose to say identity matrix, not matrix.

A. True because two matrices will have the same solution set. Ax=0 is the trivial solution, when augmenting the identity matrix with zeros the solution set will be equal to that of Ax=b.
What does Ax=b have to do with anything here? Also, x=0 is the trivial solution; Ax=0 could have more than x=0 as a solution depending on A.

B.False because we could have three independent vectors in R^3, although only two of them span a plane in R^3 while the other vector depends on one of the two. This is the reason I said false.
I'm not following your logic. You seem to be contradicting yourself. You say you have three independent vectors but only two are independent.

C.True because an nxn matrix has rows=columns, thus there must be a solution for each b in R^n.
I don't see how this follows.

D. True because the non trivial solution has a linear dependence relation. For example, let say we have a 3x4 matrix row 3 would have all zeros meaning x_3=x_3 (free) which depends on one of the other two vectors. Therefore the only two pivot positions will be in row 1 and row 2.
So is A supposed to be square or not? It seems you're not supposed to assume A is square, which affects the answers to the previous parts.

E. True because A has to be an invertible matrix to make A^T to be invertible. (A^-1)^-1=A
You're just asserting what you're supposed to prove. How does A being invertible imply A^T is invertible?

ME_student said:
The contrapositive? I have never heard of that before in my class.

For the logical statement "If P, then Q." The contrapositive is "If not Q, then not P." The contrapositive is logically equivalent to the statement.

For question E: "If At is invertible, then A is invertible." The contrapositive of this is "If A is not invertible, then At is not invertible." As you can see, the contrapositive is much easier to work with.

## 1. What is an identity matrix in linear algebra?

An identity matrix is a square matrix with 1s on the diagonal and 0s everywhere else. It is denoted by the symbol I and has the property that when multiplied by any other matrix, the resulting matrix is the same as the original matrix.

## 2. Why is the identity matrix important in linear algebra?

The identity matrix serves as the equivalent of the number 1 in regular algebra. It is used to represent the multiplicative identity element in matrix multiplication and is also used in many operations such as finding inverses and solving systems of equations.

## 3. How do you create an identity matrix?

To create an identity matrix, you simply need to have the same number of rows and columns and fill the diagonal with 1s. The remaining elements should be filled with 0s. For example, a 3x3 identity matrix would look like this:

[1 0 0]
[0 1 0]
[0 0 1]

## 4. What is the inverse of an identity matrix?

The inverse of an identity matrix is itself. This is because when an identity matrix is multiplied by its inverse, the resulting matrix is the original identity matrix.

## 5. Can an identity matrix be used in any type of matrix operation?

Yes, an identity matrix can be used in many different types of matrix operations, such as multiplication, addition, and subtraction. It is a versatile tool in linear algebra and is often used in solving problems and proving theorems.

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