[Linear Algebra] Linear Transformations, Kernels and Ranges

In summary, the linear transformations a) and b) are linear, with kernel equal to the zero vector and range equal to the span of (1,1) and (1,0) for a) and the span of (1,1) and (0,1) for b). The transformation c) is not linear, with kernel equal to the span of (0,1,0) and the range not being linear. d) is linear, with kernel equal to the set of continuous functions f such that f(a) = 0, and the range
  • #1
iJake
41
0

Homework Statement



Prove whether or not the following linear transformations are, in fact, linear. Find their kernel and range.

a) ## T : ℝ → ℝ^2, T(x) = (x,x)##
b) ##T : ℝ^3 → ℝ^2, T(x,y,z) = (y-x,z+y)##
c) ##T : ℝ^3 → ℝ^3, T(x,y,z) = (x^2, x, z-x) ##
d) ## T: C[a,b] → ℝ, T(f) = f(a)##
e) ## T: C[a,b] → C[a,b], T(f) = f^2##

Homework Equations


[/B]
Transformations are linear if ##T(a+b) = T(a) + T(b)## and if ##T(c \cdot a) = c \cdot T(a)##
##Ker(T) = \{T(x) = 0\}##
##Im(T) = \{T(x) \in W | x \in V\}##

The Attempt at a Solution


[/B]
a)
##T((x,y,z) + (a,b,c)) = ((y-x) + (b-a), (z+y) + (c+b)) = (y-x, z+y) + (b-a, c+b) = T(x,y,z) + T(a,b,c)##
##T(c \cdot (x,y,z)) = T((cx, cy, cz)) = (cy-cx, cz+cy) = (c \cdot (y-x), c \cdot (z+y)) = c \cdot (y-x, z+y) = c\cdot T(x,y,z)##

T is linear.

##Ker(T) = 0##
##Im(T) = \{(a,a) | a \in ℝ\} = <(1,1)>##

b)
##T((x,y,z) + (a,b,c)) = ((y-x) + (b-a), (z+y)+(c+b)) = (y-x, z+y) + (b-a, c+b) = T(x,y,z) + T(a,b,c)##
##T(c \cdot (x,y,z)) = T((cx,cy,cz)) = (cy-cx, cz+cy) = (c \cdot (y-x), c \cdot (z+y)) = c \cdot (y-x, z+y) = c \cdot T(x,y,z)##

T is linear.

I will try to save some space.

##Ker(T) = \{(x,y,z) \in \mathbb R^3 | T(x,y,z) = (0,0,0)\}##
##Ker(T) = \{(y,y,-y) | y \in ℝ\} = <(1,1,-1)>##
##Im(T) = y(1,1) + x(1,0) + z(0,1) = <(1,1), (1,0), (0,1)>##

c)
##T((x,y,z) + (a,b,c))## holds, but ##T(c \cdot (x,y,z))## does not hold. I am not sure if the correct notation would be that it works out to ##c^2 \cdot T(x) + c \cdot T(y,z)## but in any case it works out to a non-linear transformation.

##Ker(T) = \{(x,y,z) | (x^2, x, z-x) = (0,0,0)\}##
##(x,y,z) = (0,y,0) | y \in ℝ##
##Ker(T) = \{(0,y,0) | y \in ℝ\} = <(0,1,0)>##

##Im(T)## is not linear.

d)
##T(f+g) = (f(a) + g(a)) = f(a) + g(a) = T(f) + T(g)##
##T(c \cdot f) = (c \cdot f)(a) = c \cdot f(a) = c \cdot T(f)##

T is linear.

##Ker(T) = \{f \in C[a,b] | T(f) = 0, f \in C[a,b] | f(a) = 0\}##
##Im(T) = \{r \in ℝ | r = f(a), f \in C[a,b]\}##

e)
##T(f+g) = (f^2 + g^2) = f^2 + g^2 = T(f) + T(g)##
##T(c \cdot f) = (c \cdot f)^2 = c^2 \cdot f^2 = c^2 \cdot T(f)##

T is not linear.
##Ker(T) = \{f \in C[a,b] | f^2 = 0\}##
##Im(T)## is not linear.
 
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  • #2
look, when you see squared powers, think non linear; linear means first order. so c and e are immediately out. the others look ok.
 

Related to [Linear Algebra] Linear Transformations, Kernels and Ranges

1. What is a linear transformation?

A linear transformation is a function that maps one vector space to another in a way that preserves the structure of the vector space. It is represented by a matrix and follows the properties of linearity, such as preserving addition and scalar multiplication.

2. What is the kernel of a linear transformation?

The kernel of a linear transformation is the set of all vectors in the domain that are mapped to the zero vector in the codomain. In other words, it is the set of all vectors that the linear transformation "collapses" to zero. It is also known as the null space of the linear transformation.

3. How is the range of a linear transformation defined?

The range of a linear transformation is the set of all possible output values that the transformation can produce. It is also known as the image of the linear transformation and is a subset of the codomain.

4. What is the relationship between the kernel and range of a linear transformation?

The kernel and range of a linear transformation are related by the rank-nullity theorem. This theorem states that the dimension of the kernel plus the dimension of the range is equal to the dimension of the domain. In other words, the dimension of the kernel and range combined is equal to the total number of inputs that the transformation can "collapse" to zero.

5. How are linear transformations used in real-world applications?

Linear transformations are used in a variety of fields, including computer graphics, data analysis, and physics. They are essential for tasks such as image and signal processing, data compression, and solving systems of linear equations. They are also used in machine learning algorithms, such as linear regression and principal component analysis.

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