Let T be the linear transformation from the space of degree three or less polynomials to itself defined by the following formula: T(ax + bx²+cx+ d) = (a - b)x³ + (a - b)x² + (2c+ d)x+(2c+d) Find a basis for the null space of T.

Q: Find 7A. 7A = A = 4 7 0 -3 00

A: To find the value of matrix after the multiplication of 7 on them.

Q: Show Mat Res (r(z), -n) (1)" n!

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Q: Differentiate .3 to 1-x3 w.r.t. x3.

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Q: Find an explicit formula for 1/(2-1/2), 2/(3-1/3),3/(4-1/4),4/(5-1/5)...

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Q: Let A = 12 3 14 0 1 02 -3 0 1000 What are the rank and nullity of A?

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Q: -1 Let A = 01. Compute A100 -2 1 1 3.

A: To compute the A100 first we need to diagonalize it to make the computation very easy.

Q: Prove (1.2), (1.3), and (1.4).  

A: (a) Let x∈A∪B So, ⇔x∈A or x∈B⇔x∈B or x∈A⇔x∈B∪A So, A∪B=B∪A

Q: 2 find the der ivative d.

A: Consider a function is defined as, y=fx·gx The derivative of such a function is determined using…