Let F be a fixed 3x2 matrix, and let H be the set of all matrices A in M_(2×4) with the property that FA = 0 (the zero matrix in M_(3×4)). Determine if H is a subspace of M_(2×4)

asked 2021-02-11
Let F be a fixed 3x2 matrix, and let H be the set of all matrices A in \(\displaystyle{M}_{{{2}×{4}}}\) with the property that FA = 0 (the zero matrix in \(\displaystyle{M}_{{{3}×{4}}}{)}\). Determine if H is a subspace of \(\displaystyle{M}_{{{2}×{4}}}\)

Answers (1)

A subspace of a vector space V is a subset H of V with 3 properties:
1. the zero vector of V is in H
2. the subspace is closed under addition
3. the subspace is closed under scalar multiplication.
1. Let A = 0. Then for any matrix F, FA = 0. Thus, \(\displaystyle{A}\in{H}\). So H contains the zero vector.
2. Let \(\displaystyle{A}_{{1}},{A}_{{2}}\in{H}\). Consider \(\displaystyle{A}_{{1}}+{A}_{{2}}.{F}{\left({A}_{{1}}+{A}_{{2}}\right)}={F}{A}_{{1}}{4}+{F}{A}_{{2}}\) due to the distributive property of matrices. Since \(\displaystyle{A}_{{1}},{A}_{{2}}\in{H},{F}{A}_{{1}}+{F}{A}_{{2}}={0}+{0}={0}\).
Thus, \(\displaystyle{A}_{{1}}+{A}_{{2}}\in{H}\), so H is closed under addition.
3. Let \(\displaystyle{c}\in{R},{A}\in{H}\). Consider cA. F(cA) = c(FA) according to scalar properties. Since \(\displaystyle{A}\in{H}\), c(FA) =c(0) =0. Thus, \(\displaystyle{c}{A}\in{H}\). So H is closed under scalar multiplication.
Thus, H fulfills all the requirements of the definition of a subspace of \(\displaystyle{M}_{{{2}{x}{4}}}\).

Relevant Questions

asked 2021-02-08
Let \(M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})\) be the set of 2 x 2 matrices with the entries in \(\mathbb{Z}/\mathbb{6Z}\)
a) Can you find a matrix \(M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})\) whose determinant is non-zero and yet is not invertible?
b) Does the set of invertible matrices in \(M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})\) form a group?
asked 2021-01-28
Let W be the subspace of all diagonal matrices in \(M_{2,2}\). Find a bais for W. Then give the dimension of W.
If you need to enter a matrix as part of your answer , write each row as a vector.For example , write the matrix
asked 2020-11-10
Determine whether the subset of \(M_{n,n}\) is a subspace of \(M_{n,n}\) with the standard operations. Justify your answer.
The set of all \(n \times n\) matrices whose entries sum to zero
asked 2021-01-04
In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.
asked 2020-11-01
Give a full correct answer for given question 1- Let W be the set of all polynomials \(\displaystyle{a}+{b}{t}+{c}{t}^{{2}}\in{P}_{{{2}}}\) such that \(\displaystyle{a}+{b}+{c}={0}\) Show that W is a subspace of \(\displaystyle{P}_{{{2}}},\) find a basis for W, and then find dim(W) 2 - Find two different bases of \(\displaystyle{R}^{{{2}}}\) so that the coordinates of \(\displaystyle{b}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{5}\backslash{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\) are both (2,1) in the coordinate system defined by these two bases
asked 2021-02-02
Solve the problem. Let vector u have initial point P1=(0,2) and terminal point P2=(−2,6). Let vector v have initial point Q1=(3,0) and terminal point Q2=(1,4). U and v have the same direction. Find ||u|| and ||v||. Is u=v?
asked 2020-10-21
This is the quesetion. Suppose that a does not equal 0.
a. if \(\displaystyle{a}\cdot{b}={a}\cdot{c}\), does it follow that b=c?
b. if \(\displaystyle{a}\times{b}={a}\times{c}\), does it follow that b=c ?
c. if \(\displaystyle{a}\cdot{b}={a}\cdot{c}\) and \(\displaystyle{a}\times{b}={a}\times{c}\), does it follow that b=c?
Either prove the assertion is true in general or show that it is false for a concret choice of vectors a, b, c
asked 2020-10-23
1. Find each of the requested values for a population with a mean of \(? = 40\), and a standard deviation of \(? = 8\) A. What is the z-score corresponding to \(X = 52?\) B. What is the X value corresponding to \(z = - 0.50?\) C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of \(M=42\) for a sample of \(n = 4\) scores? E. What is the z-scores corresponding to a sample mean of \(M= 42\) for a sample of \(n = 6\) scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: \(a. -2.00 b. 1.25 c. 3.50 d. -0.34\) 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with \(\mu = 78\) and \(\sigma = 12\). Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: \(82, 74, 62, 68, 79, 94, 90, 81, 80\). 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about \($12 (\mu = 42, \sigma = 12)\). You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is $44.50 from tips. Test for a difference between this value and the population mean at the \(\alpha = 0.05\) level of significance.
asked 2021-01-30
Given the matrices
\(A=\begin{bmatrix}5 & 3 \\ -3 & -1 \\ -2 & -5 \end{bmatrix} \text{ and } B=\begin{bmatrix}0 & -2 \\ 1 & 3 \\ 4 & -3 \end{bmatrix}\)
find the 3x2 matrix X that is a solution of the equation. 2X-A=X+B
asked 2020-12-30
Show that C'[a, b] is subspace of C [a, b]