Find the dimensions of the following vector spaces.(a) The vector space R5(b) The vector space M6×7(c) The vector space of 2 × 2 matrices with trace 0(d) The vector space P3 of polynomials with degree less than or equal to 3(e) The vector space of all diagonal 2 × 2 matrices(f) The vector space of all lower triangular 4 × 4 matrices Find a basis {p(x), q(x)} for the vector space {f(x) = P2 | ƒ'(−8) = f(1)} where P₂ is the vector space of polynomials in x with degree less thanor equal to 2.p(x) = ☐, q(x) == 0

(a) The vector space R5 The dimension of R5 is simply the number of components in each vector.Since…

Answered: Find the dimensions of the following…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.6: Rank Of A Matrix And Systems Of Linear Equations

Problem 69E: Consider an mn matrix A and an np matrix B. Show that the row vectors of AB are in the row space of...

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Consider an mn matrix A and an np matrix B. Show that the row vectors of AB are in the row space of B and the column vectors of AB are in the column space of A.
Find the dimension of the following vector spaces. (a) V = {0}: dim V = (b) V = R¹ : dim V= (c) V is the vector space containing all polynomials: dim V= (d) V is the vector space containing all 2 x 3 matrices: dim V- (enter integers; if V is infinite-dimensional, enter infinity in lower case)
The set of all polynomials of degree 6 under the standard addition and scalar multiplication operations is not a vector space because We can find a polynomial P(x) such that (c+d)P(x)=cP(x)+dP(x). We can find two polynomials P(x) O and Q(x) for which P(x)·Q(x)=Q(x)·P(x) We can find a polynomial P(x) for which 1-P(x)=P(x) It is not closed under addition.
Find the dimmensions of the following vector spaces
Find the dimensions of the following vector spaces. (a) The vector space of 2 x 2 matrices with trace 0 (b) The vector space P3[x] of polynomials with degree less than 3 (c) The vector space of all diagonal 2 x 2 matrices (d) The vector space of all lower triangular 6 x 6 matrices (e) The vector space RX (1) The vector space R 4
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Find a formula for the dimension of the vector space of symmetric n X n matrices.
The set of all polynomials of degree 7 under the standard addition and scalar multiplication operations is not a vector space because * O We can find a polynomial P(x) such that (c+d)P(x)#cP(x)+dP(x). OWe can find a polynomial P(x) for which 1 P(x)#P(x) We can find two polynomials P(x) and Q(x) for which P(x)-Q(x)#Q(x)-P(x) It is not closed under addition.
The set of all polynomials of degree 4 under the standard addition and scalar multiplication operations is not a vector space because * O We can find two polynomials P(x) and Q(x) for which P(x)-Q(x)#Q(x) P(x) O It is not closed under addition. O We can find a polynomial P(x) for which 1-P(x)#P(x) O We can find a polynomial P(x) such that (c+d)P(x)#cP(x)+dP(x).
Find the dimensions of the following linear spaces. (a) R2x7 (b) The space of all diagonal 6 x 6 matrices (c) The real linear space C5
Find the examples that satisfy the statement in the given items and show some work how they are correct examples. (a) A 4 × 4 matrices A that is not digonalizable.(b) Two matrices A and B that have the same characteristic polynomial but are not similar.(c) Two diagonal matrices D1 and D2 that are similar but they are not equal.(d) Two similar matrices A and B such that A is orthogonally digonalizable but B is not orthogonally diagonalizable.
Which of the following sets are closed under scalar multiplication? (i) The set of all vectors in R of the form (a, b) where a + 2b = 0. (ii) The set of all 3 × 3 matrices whose trace is equal to 0. (iii) The set of all polynomials in P, of the form ao + aj x+ az x² where the product ao a1 az = 0.

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