Linear algebra proof with operator T^2 = cT

In summary, the operator T satisfies T^2=cT with c ≠ 0 and the vector space V can be decomposed into a direct sum of the eigenspace U and the kernel of T, where U is the set of vectors u satisfying T(u)=cu with c as the eigenvalue. Since T^2=cT, c cannot be equal to 0, otherwise we would have T(0).
  • #1
evilpostingmong
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0

Homework Statement


Let T:V--->V be an operator satisfying T^2=cT c=/=0.
Show that V=U[tex]\oplus[/tex]kerT U={u l T(u)=cu}

Homework Equations


The Attempt at a Solution


Now before I start, just one quick question about ker T:
U seems to be an eigenspace since T(u)=cu with c the eigenvalue.
But that must mean that the kernel has 0 as the only element since
0 is not an eigenvector so it can't be in the eigenspace, right?
And, since T^2=cT, c cannot be 0 otherwise we would have T(0).
 
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  • #2
Hi evilpostingmong! :smile:

(have a not-equal: ≠ and try using the X2 tag just above the Reply box :wink:)
evilpostingmong said:
Let T:V--->V be an operator satisfying T^2=cT c=/=0.
Show that V=U[tex]\oplus[/tex]kerT U={u l T(u)=cu}

Now before I start, just one quick question about ker T:
U seems to be an eigenspace since T(u)=cu with c the eigenvalue.
But that must mean that the kernel has 0 as the only element since
0 is not an eigenvector so it can't be in the eigenspace, right?
And, since T^2=cT, c cannot be 0 otherwise we would have T(0).

T2 = cT means T is a projection …

so suppose eg V is R3, and T is the projection onto the x-y plane (with c = 1),

then kerT is the z-axis. :wink:
 

1. What is a linear algebra proof?

A linear algebra proof is a logical argument that uses mathematical principles to demonstrate the validity of a statement or theorem in the field of linear algebra. It involves using properties and operations of vectors and matrices to arrive at a conclusion.

2. What does it mean for an operator T to have T^2 = cT?

When an operator T satisfies the equation T^2 = cT, it means that applying the operator T twice is equivalent to multiplying it by a scalar c. This is known as the eigenvalue equation, and c is the eigenvalue associated with T.

3. How is the eigenvalue c related to the properties of operator T?

The eigenvalue c is related to the properties of operator T in the following ways:

  • If c = 0, then T is a zero operator, meaning it maps all vectors to the zero vector.
  • If c = 1, then T is an identity operator, meaning it leaves all vectors unchanged.
  • If c is a non-zero constant, then T is a scalar multiple of the identity operator.
  • If c is a negative constant, then T is a reflection operator.

4. How can I prove that an operator T satisfies the equation T^2 = cT?

To prove that an operator T satisfies the equation T^2 = cT, you can use the properties of linear operators and the definition of eigenvalues. Start by showing that T^2 = cT for a specific vector, and then use the linearity property of T to extend it to all vectors. You can also use the fact that c is an eigenvalue to show that T^2 = cT.

5. What are the applications of T^2 = cT in linear algebra?

The equation T^2 = cT is used in various applications in linear algebra, such as finding eigenvalues and eigenvectors, diagonalizing matrices, and solving differential equations. It is also used in the study of linear transformations, where T represents a transformation and T^2 represents the composition of two transformations.

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