Linear independence and dependence

In summary, linear independence refers to a set of unique vectors that cannot be created by adding or multiplying each other, while linear dependence refers to a set of vectors where at least one vector can be written as a linear combination of the others. To determine if a set of vectors is independent or dependent, we can use the concept of a linear combination. Linear independence is important in linear algebra as it allows us to understand the structure and behavior of vector spaces. In a three-dimensional space, any set of more than three vectors is guaranteed to be linearly dependent. This concept can also be used to solve systems of linear equations by reducing the number of variables.
  • #1
yoyo
21
0
Hi everyone, having problems with this question, can anyone please help

Question: consider a 2 x 2 matrix, can you construct a matrix whose columns are linearly dependent and whose rows are linearly independent?

My answer is no. I cannot think of any combination that would make this true? Or is there? If so why?

Thanks
 
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  • #2
No, because having linearly dependent columns gives a singular matrix yet linearly independent rows gives a non-singular matrix.
 
  • #3
Thanks, make sense.
 
  • #4
just do the algebra.
 

1. What is the difference between linear independence and linear dependence?

Linear independence refers to a set of vectors in which no vector can be written as a linear combination of the other vectors. In other words, the vectors are all unique and cannot be created by adding or multiplying each other. On the other hand, linear dependence refers to a set of vectors in which at least one vector can be written as a linear combination of the other vectors. This means that some vectors in the set are redundant and can be created by combining other vectors.

2. How do you determine if a set of vectors is linearly independent or dependent?

To determine if a set of vectors is linearly independent or dependent, we can use the concept of a linear combination. If we can find a non-zero solution to the equation c1v1 + c2v2 + ... + cnvn = 0, where c1, c2, ..., cn are constants and v1, v2, ..., vn are the vectors in the set, then the vectors are linearly dependent. If the only solution to the equation is c1 = c2 = ... = cn = 0, then the vectors are linearly independent.

3. Why is linear independence important in linear algebra?

Linear independence is important in linear algebra because it allows us to understand the structure and behavior of sets of vectors. Linearly independent vectors are essential in creating a basis for a vector space, which is a set of vectors that can be used to represent any other vector in that space. This is important in many applications, such as data analysis, computer graphics, and machine learning.

4. Can a set of two vectors be linearly dependent in a three-dimensional space?

Yes, a set of two vectors can be linearly dependent in a three-dimensional space. In fact, any set of more than three vectors in a three-dimensional space is guaranteed to be linearly dependent. This is because a three-dimensional space has three dimensions, so any set of more than three vectors will have at least one redundant vector that can be created by combining the others.

5. How can we use linear dependence to solve systems of linear equations?

In a system of linear equations, each equation can be represented as a linear combination of the variables. If the system has more variables than equations, it is considered an overdetermined system and is likely to have infinitely many solutions. In this case, we can use linear dependence to find the variables that create a linear combination of the other variables, and thus, reduce the number of variables in the system. This makes it easier to solve for a unique solution.

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