Linear Algebra Null Space and Range

• Punkyc7
In summary, the conversation is discussing the range and null space of a transformation T that maps polynomials of degree <= 2 to polynomials of degree <= 1. The null space is found to be spanned by the vector {1}, and there is a discrepancy in the calculation of the range. While one person believes the range is {x, x^2}, another person argues that it should be {1, x} based on the given information. It is clarified that the null space and range are in different spaces and have no relation to each other, and the correct calculation for the range is shown to be {2ax + (6a + b)}. The given transformation is 3p'' - p', not p''
Punkyc7
give a basis for the range and the null space of T:P2(R) to P1(R)
where for all p element of P2(R), T(p)=3p'' - p'

I got the null space is {1} and the range is {x,x^2} but the answer says it should be {1,x} for the range. How can something be apart of the null space and the range if its not the zero vector?

The range can't possibly include x2 since you are given that T maps polynomials of degree <= 2 to polynomials of degree <= 1.

If p(x) = ax2 + bx + c, then T(p) = -2ax + 6a - b. The only way T(p) = 0 for all x is if both a and b are zero.

The null space and range are in different spaces so have nothing to do with one another (except that their dimensions add to the dimension of the domain space). Yes, if y is a constant y''- y'= 0 so the null space is spanned by {1} (I would not say the null space is {1}- that's just a single vector). If y is in the range, then p''- p'= y. If $p= ax^2+ bx+ c$ then $3p''-p'= 6a+ 2ax+ b= 2ax+ (6a+b)$

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HallsofIvy said:
The null space and range are in different spaces so have nothing to do with one another (except that their dimensions add to the dimension of the domain space). Yes, if y is a constant y''- y'= 0 so the null space is spanned by {1} (I would not say the null space is {1}- that's just a single vector). If y is in the range, then p''- p'= y. If $p= ax^2+ bx+ c$ then $p''-p'= 2a+ 2ax+ b= 2ax+ (2a+b)$
It's given that T(p) = 3p'' - p', not p'' - p'. This affects the calculation of the range.

What is the null space of a matrix?

The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in a zero vector. In other words, it is the set of all solutions to the homogeneous system of equations represented by the matrix.

What is the range of a matrix?

The range of a matrix is the set of all possible output vectors that can be obtained by multiplying the matrix with any input vector. In other words, it is the set of all solutions to the non-homogeneous system of equations represented by the matrix.

How do you find the null space and range of a matrix?

To find the null space of a matrix, you can perform row reduction on the matrix and look for the row(s) that contain all zeros. The column(s) corresponding to these rows will form the basis for the null space. To find the range of a matrix, you can perform row reduction and look for the columns that contain the pivot variables. The corresponding columns in the original matrix will form the basis for the range.

What is the dimension of the null space and range of a matrix?

The dimension of the null space of a matrix is equal to the number of free variables in the row-reduced form of the matrix. The dimension of the range is equal to the number of pivot variables in the row-reduced form of the matrix.

Why are the null space and range important in linear algebra?

The null space and range provide valuable information about the solutions to a system of linear equations represented by a matrix. They also help in understanding the properties and behavior of linear transformations and can be used to solve various problems in engineering, physics, and computer science.

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