Linear Transformation Proofs

[knca987]

These proofs are stumping me. I have an idea for both but don't really know where to take them...

1) Let T (domain R[sup:24r9cjoh]n[/sup:24r9cjoh] and codomain R[sup:24r9cjoh]n[/sup:24r9cjoh]) be a linear transformation. Show that if T maps two linearly independent vectors onto a linearly dependent set, then the equation T(x)=0 has a nontrival solution. (the x and 0 are vectors). Suppose vectors u and v in R[sup:24r9cjoh]n[/sup:24r9cjoh] are linearly independent and yet T(u) and T(v) are linearly dependent. Then c[sub:24r9cjoh]1[/sub:24r9cjoh]*T(u)+c[sub:24r9cjoh]2[/sub:24r9cjoh]*T(v) = 0 for some weights c[sub:24r9cjoh]1[/sub:24r9cjoh] and c[sub:24r9cjoh]2[/sub:24r9cjoh], not both zero. Use this equation.

2) Suppose vectors v[sub:24r9cjoh]1[/sub:24r9cjoh],...,v[sub:24r9cjoh]p[/sub:24r9cjoh] span R[sup:24r9cjoh]n[/sup:24r9cjoh] and let T with domain R[sup:24r9cjoh]n[/sup:24r9cjoh] and codomain R[sup:24r9cjoh]n[/sup:24r9cjoh] be a linear transformation. Suppose T(v[sub:24r9cjoh]i[/sub:24r9cjoh]) = 0 for i = 1,...,p. Show that T is the zero transformation. That is, show that if x is any vector in R[sup:24r9cjoh]n[/sup:24r9cjoh], then T(x) = 0. (the zero vector)

Thanks!

 

[Deleted member 4993]

These proofs are stumping me. I have an idea for both but don't really know where to take them...

1) Let T (domain R^n and codomain R^n) be a linear transformation. Show that if T maps two linearly independent vectors onto a linearly dependent set, then the equation T(x)=0 has a nontrival solution. (the x and 0 are vectors). Suppose vectors u and v in R^n are linearly independent and yet T(u) and T(v) are linearly dependent. Then c1*T(u)+c2*T(v) = 0 for some weights c1 and c2, not both zero. Use this equation.

2) Suppose vectors v1,...,vp span R^n and let T with domain R^n and codomain R^n be a linear transformation. Suppose T(v sub i) = 0 for i = 1,...,p. Show that T is the zero transformation. That is, show that if x is any vector in R^n, then T(x) = 0. (the zero vector)

Thanks!

Please share your idea with us - so that we may know where to begin to help you.

 

[knca987]

Well, my idea for the first question completely fell through, so I am not sure where to start with it. As for the second, I am pretty sure I need to utilize the property T(0) = T(0x)=0T(x)=0, with x and 0 being vectors, since we know that T is a linear transformation. But I don't think that is sufficient.