Linear Transformations, Span, and Independence

In summary, there is a linear algebra theorem that states for a linear transformation T:Rn -> Rm and its standard m x n matrix A, if the columns of A span Rn then the transformation is onto, and if the columns of A are linearly independent then the transformation is one-to-one. This theorem may not be explicitly mentioned in the textbook but could have been discussed in lecture. Additionally, if m>n, then T cannot be onto and if the columns are independent, T(Rn) will be an n-dimensional subspace of Rm.
  • #1
tandoorichicken
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0
Is there a linear algebra theorem or fact that says something like

For a linear transformation T:Rn -> Rm and its standard m x n matrix A:
(a) If the columns of A span Rn the transformation is onto.
(b) If the columns of A are linearly independent the transformation is one-to-one.

Is this correct? I can't find it anywhere in my textbook but it may have been mentioned in lecture. Any insight would be appreciated.
 
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  • #2
If m>n, then can T be onto? If the colomns are independent, T(Rn) will be an n-dimensional subspace of Rm.

b) is true, as you can easily prove yourself. Show that T(v1)=T(v2) implies that v1=v2.
Hint: If you column-vectors are independent you can use them as a basis.
 

1. What is linear independence?

Linear independence refers to a set of vectors in a vector space that cannot be written as a linear combination of each other. In other words, no vector in the set can be expressed as a combination of the other vectors in the set. This means that each vector in the set contributes a unique component to the overall span.

2. How do you determine if a set of vectors are linearly independent?

To determine if a set of vectors are linearly independent, we can use the concept of linear dependence. If a set of vectors is linearly dependent, it means that at least one vector in the set can be expressed as a linear combination of the other vectors. On the other hand, if a set of vectors is linearly independent, then no vector can be written as a linear combination of the other vectors. One way to check for linear independence is by using the determinant method, where we create a matrix with the given vectors and take the determinant. If the determinant is non-zero, the vectors are linearly independent.

3. What is the span of a set of vectors?

The span of a set of vectors is the set of all possible linear combinations of those vectors. In other words, the span is the set of all vectors that can be created by scaling each vector in the set and adding them together. The span can be thought of as the "space" that the set of vectors can fill.

4. How do you find the span of a set of vectors?

To find the span of a set of vectors, we can use the linear combination method. This involves taking each vector in the set and multiplying it by a scalar, then adding all the resulting vectors together. The resulting vector is then added to the span. This process is repeated for every vector in the set, and the resulting span is the set of all possible linear combinations of the original vectors.

5. Why is linear independence important in linear algebra?

Linear independence is important in linear algebra because it allows us to understand the relationship between vectors in a vector space. It helps us determine if a set of vectors is a basis for the vector space, which is essential in solving systems of linear equations. It also helps us understand the dimension of a vector space, and how many independent vectors are needed to span the entire space.

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