Suppose A is 10 by 10 and A2 = 0 (zero matrix). So A multiplies each column of A to give the zero vector. This means that the column space of A is contained in the __ . If A has rank r, those subspaces have dimension r � 10 � r. So the rank is '(' � 5.

Q: 1. Is W=set of upper triangular 3 x 3 matrices a subspace of V=all 3 x 3 matrices? Support your…

A: I have showed that any two vectors form W linear combination of these also lies on W

Q: Find the orthogonal projection of 9. -3 = 4 onto the subspace W of Rª spanned by -3 -4 3 and 6 -3…

A: Solution:-

Q: Suppose U and W are subspaces of V such that dimU=4, dimW=5, and dimV=7. Find the possible…

A:

Q: 1

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Q: Find a basis for the subspace of R4 spanned by S. S = {(2, 5, –3, -2), (-2, –3, 2, -5), (1, 3, –2,…

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Q: What is the dimension of the subspace or the space of 5 x 5 matrices (ai) sucn that a11 +...+ az5 =…

A:

Q: Is the set of all 2 x 2 matrices of the form a 1/ 1 b , where a and b may be any scalars, a vector…

A: The given set is a11b | a, b∈K. It is known that any subspace should contain the identity element.

Q: Find the standard matrix for the orthogonal projection of R3 onto the subspace spanned by aj=(1,1)…

A: The vectors that span the subspace are a1=1, 1 and a2=0, 1. The normalising vectors are 12, 12 and…

Q: 2. How can you build a subspace of a vector space V by using a single vector vEV

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Q: The set of all nxn matrices whose entries are negative or zero is a subspace of Mnn: True False

A: The answer is true

Q: The set of all 3 x 3 matrices of the form 0 0 0 0 0 -5x 0 0 0 where x is a real number, is a…

A: Given :M=00000000-5x

Q: Show that the set of all solutions to the equation Ax = 0, where   A = [1 2 3 -2 -1 0]         [1 2…

A:

Q: Find a basis for the subspace of R4 spanned by S.S = { (6, −3, 6, 34), (3, −2, 3, 19), (8, 3, −9,…

A: Given: S = { (6, −3, 6, 34), (3, −2, 3, 19), (8, 3, −9, 6), (−2, 0, 6, −5)} To find: Basis spanned…

Q: Find the subspace spanned by the three vectors [2 3 1]T, [2 1-5]T, and [2 4 4]T Feuth

A: We will check the  vectors that it is linearly dependent or linearly independent.  If the vectors…

Q: Find a basis for the subspace of R* spanned by S. S = {(-20, –7, 2, –4), (21, 7, –2, 4), (-9, –3, 1,…

A: We have given set of vectors ,  S = -20 , -7 , 2 , -4 , 21 , 7 , -2 , 4 , -9 , -3 , 1 , -2 , -18 ,…

Q: What can be said about Nul A if A is a 5×7 ​matrix? Could it be the zero​ subspace? Why or why​ not?

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Q: Let H be the collection of all 2 × 4 matrices whose second and fourth columns are equal. Show that H…

A:

Q: 3. Without computing A, find bases for the four fundamental subspaces if: [1 0 0 6 1 0 9 8 [1 2 3 4…

A: Answer is mentioned below

Q: Find a basis for the subspace of R* spanned by S. S = {(43, -21, 6, 12), (-14, 7, -2, –4), (6, –3,…

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Q: Find a basis for the subspace of R4 spanned by S. S = {(2, 9, -2, 53), (-4, 2, 4, –2), (8, –4, –8,…

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Q: 5. The solution set of the system Ax 0 with three unknowns is a subspace of the vector space M33 O…

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Q: Find a basis for the subspace of R4 spanned by S. S = {(2, 9, -2, 53), (-5, 2, 5, –2), (8, –5, –8,…

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Q: Find a basis for the subspace of R4 spanned by S. S = {(-11, 6, 4, 8), (-6, 3, 2, 4), (-2, 1, 1, 2),…

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Q: Show that the solution vectors of a consistent non-homogeneous system of m equations in n unknowns…

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Q: Let A be a 80 x 90 matrix (80 rows and 90 columns). Does the homogeneous 0 always has nonzero…

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Q: The set of all nxn matrices A such that the linear system Ax = 0 has only trivial solutions are the…

A:

Q: Is W=set of upper triangular 3 x 3 matrices a subspace of V=all 3 x 3 matrices?

A: In this question we need to check W is a subspace. so we need to check two properties * w1 and w2…

Q: Find a basis for the subspace of R4 spanned by S. S = {(3, 9, –3, 53), (-4, 3, 4, –3), (8, –4, –8,…

A:

Q: Find a basis for the subspace spanned by the given vectors. 3 Span 3 -1,2 1 3 4

A: The given subspace consist of all vectors in Span-231, 3-12, 123, -154. Take S=-231, 3-12, 123,…

Q: The set of all nxn matrices whose entries are odd integers is a subspace of Mnn. Is it true or…

A: A subset of a vector space is a subspace if it is a vector space itself. First of all the space…

Q: Find a basis for the subspace of R4 spanned by S. S = {(3, 1, 1, 3), (-2, –1, -1, -3), (4, 2, 1, 3),…

A: Let S=3, 1, 1, 3, -2, -1, -1, -3, 4, 2, 1, 3, -12, -6, -3, -8

Q: (a) If a 7 by 9 matrix has rank 5, what are the dimensions of the four subspaces? What is the sum of…

A: Use the given information.

Q: 2. Show that the set of all diagonal matrices of order 2 x 2 is a subspace of M2x2. That [ ] a is,…

A:

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A:

Q: Let A be a 7 × 5 matrix with rank equal to 4 andlet b be a vector in R8. The four fundamental…

A: For matrix A=mxn                   = 7x5Rank of A=4Now ATA is a 5x5 matrix ∵ Null ATA=NullA⇒dim…

Q: The set of all 3 x 3 matrices of the form 1 0 0 0 1 0 00 -a Select one: True O False + 13x3 where a…

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Q: 3 3 -1 A=| 1 3 1

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Q: The set of all solutions of homogeneous system AX = 0, where A is a 4 × 3 matrix, is a subspace of…

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Q: Find a basis for the subspace of R4 spanned by S. S = {(2, 5, –3, -2), (-2, –3, 2, –5), (1, 3, -2,…

A:

Q: Find a basis for the subspace of R4 spanned by S. S = {(3, -2, -3, 6), (2, –2, –3, 6), (-1, 1, 1,…

A:

Q: Find the orthogonal projection of onto the subspace W of Rª spanned by projw() = v= 4 -2 -13] 3 15…

A:

Q: Find a basis for the subspace of R4 spanned by S. S= {(5, 9, -5, 53), (-2, 5, 2, -5), (8, -2, -8,…

A:

Q: Let A be a 3 × 2 matrix with rank 2. Give geometric descriptions of R(A) and N(AT ), and describe…

A: Given: Rank of matrix A3x2 is 2. Then the dimension of column matrix A dimR(A)=2 Also given that the…

Q: Suppose that A is an nx m matrix of rank 3, the nullity of A is 4, and the column space of A is a…

A: Solution:-

Q: The set of all nxn matrices having trace equal to zero. is a subspace W of Mnxn(F). Find a basis for…

A:

Q: 2. Find a 3 x 4 matrix R for which the null space of R is spanned by the vectors -- -E 0. v = u =…

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Q: Let A e Mnxn and W be the set of all solutions for the homogenous linear system AX = 0. Prove or…

A: Given, A∈Mn×n and W be the set of all solutions for the homogenous linear system AX=0. We have to…

Q: If A has the same four fundamental subspaces as B, does A = B?

A: Four Fundamental Subspaces: Let be a matrix of order and be its transpose matrix. Then The row…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.