TBA 35 Hi guys and gals I need help with these Linear Algebr

TBA 3-5 ++++++++++++++Hi guys and gals. I need help with these Linear Algebra questions. They are true or false, and I would like to get an opinion from an expert. Thank you. A clear and correct answer will get a thumbs up guaranteed. Thank you!!!!!!!!!!!!!!++++++++++++++++++

Mark each statement true or false and justify your answer No justification, no credit. For questions that are not true/false, provide clear and complete solutions. A linear transformation T : Rn Rm always maps the origin of R\" to the origin of R\" 3. 4. Define,f : R R by/(x) = mx + b. a. Show that/is a linear transformation when b = 0. b. Find a property of a linear transformation that is violated when b 0. 5. Show that the transformation T defined by 2242) is not linear

Solution

3) This is true. We know that we can write this kind of linear transformation as y=Ax where x is in R^n and y is R^m. Therefore, whenver we have x=Origin in R^n, we will get y as product of A and x, which will eventually be the zero vector in R^m. Hence any linear transformation will map the origin of input space to origin of output space.

4)

a) For f(x) = mx+b to be linear transformation, it must be closed under scalar multiplication.

That is, f(cx) = cf(x)

f(cx) = m(cx)+b = mcx+b = c(mx+b/c)

f(cx) is only equal to cf(x) when b=0, therefore, at b=0, the given transformation is closed under scalar multiplication. And it can be easily shown that this transformation is closed under vector addition. Therefore, at b=0, given transformation is linear transformation.

(b) The property which is voilation at b!=0 is \'Closeness under scalar multilication\'

5) Given T(x₁,x₂) = (x₁-2|x₂|,x₁-4x₂)

The given transformation is not closed under vector addition, because |x1|+|x2| is not necessarily equal to |x1+x2|, therefore, this transformation is not linear.