trickslapper
Junior Member
- Joined
- Sep 17, 2010
- Messages
- 62
How can i find out if a vector can be expressed as a linear combination of other vectors using rank? I tried to follow the example in my book and i am just not getting it, this is what my book says:
Can the vector [1 1] be written as a linear combination of the vectors [3 6] and [2 4]?
Solution: (I'm going to put the main things the book says)
A can be row reduced to: [3 6; 0 0]. First of all i'm pretty sure thats wrong but whatever. The book goes on to say that this has rank one, ok i can buy that.
Next the book says "In contrast, the matrix B=[1 1; 3 6; 2 4] can be row reduced to: [1 1; 0 1; 0 0].
"which has rank two; hence B has two linearly independent row vectors. Since B is precisely A withe one additional row, it follows that the additional row [1 1] is independent of the other two and, therefore, cannot be written as a linear combination of the other two vectors."
Ok so i tried this same process on some homework problems and i got the wrong answer. Basically can someone show me how to find out if a vector can be expressed as a linear combination of some other vectors using rank? I know sometimes its kind of obvious but i don't get what my book is trying to say and i need to know how to solve these kinds of problems.
EDIT: After working it out some more, i found a way to get the right answer. Basically i put the vectors given to me in a matrix and augment it with the vector that i'm trying to decide is a lin combination of the others or not. Once i row reduce, if i can find a solution then yes the vector can be written as a linear combination. I'm pretty sure this isn't what the book had in mind but its how i'm doing it on my test.
Can the vector [1 1] be written as a linear combination of the vectors [3 6] and [2 4]?
Solution: (I'm going to put the main things the book says)
A can be row reduced to: [3 6; 0 0]. First of all i'm pretty sure thats wrong but whatever. The book goes on to say that this has rank one, ok i can buy that.
Next the book says "In contrast, the matrix B=[1 1; 3 6; 2 4] can be row reduced to: [1 1; 0 1; 0 0].
"which has rank two; hence B has two linearly independent row vectors. Since B is precisely A withe one additional row, it follows that the additional row [1 1] is independent of the other two and, therefore, cannot be written as a linear combination of the other two vectors."
Ok so i tried this same process on some homework problems and i got the wrong answer. Basically can someone show me how to find out if a vector can be expressed as a linear combination of some other vectors using rank? I know sometimes its kind of obvious but i don't get what my book is trying to say and i need to know how to solve these kinds of problems.
EDIT: After working it out some more, i found a way to get the right answer. Basically i put the vectors given to me in a matrix and augment it with the vector that i'm trying to decide is a lin combination of the others or not. Once i row reduce, if i can find a solution then yes the vector can be written as a linear combination. I'm pretty sure this isn't what the book had in mind but its how i'm doing it on my test.