Basis of a Vector Space and its Subspace 

DefinitionLet $V$ is a vector space over a field $F$ and $S$ is a nonempty subset of $V$.
We say that W is a Subspace of $V$ if $S$ is also a vector space over $F$ under the operations of $V$. Equivalently, $W$ can also be a subspace of $V$ whenever $w_1, w_2 \in$ W and $c_1, c_2 \in F$, it follows that $c_1w_1$ + $c_2w_2 \in W$. A subset $B$ of $V$ is called a Basis for $V$ if $B$ is linearly independent over $F$ and every element in $V$ is a linear combination of elements of $B$. 8 
MotivationThe subspace in linear algebra is analogous to the idea of subsets. A subset is a set of which all the elements are contained in another set. Consider a vector space $V$=$\mathbb{R}^2$ and take any vector, say (2, 5) in the space. Now, we know that this vector can be expressed as 9 
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Figure 1: Basis of a 2D Vector Space 
Bird's Eye ViewConsider a vector space $V$ such that $X, Y$ are its subspaces. The subset $X \cap Y$ contains a zero vector and is closed under addition and scalar multiplication. Moreover, if $X_1,X_2, . . . . . . ,X_n$ are the subspaces of a vector space $V$, then the intersection of all the subspaces $X_1 \cap X_2 \cap . . . . .\cap X_n$ will also be a subspace. 11 
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Figure 2: Intersection of two subspaces is a subspace. 
Context of the Definition

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Figure 3: Straight lines as subspaces 
2) Let $W = \{ (x, y, z) \in \mathbb{R}^3$  $ax + by +cz = 0$, $a, b, c \in \mathbb{R}$ and are fixed$\}$ is a subspace of $\mathbb{R}^3$. 15 
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Figure 4: The planes passing through the origin are the subspaces of $\mathbb{R}^3$ but the line is not a subspace. 
Note: The set $\mathbb{R}^n$ is a subspace itself, since it contains zero and is closed under the scalar multiplication and addition. 3) Consider an ordinary differential equation $\frac{d^2y}{dx^2} + ay$ = 0, $a \in \mathbb{R}$.
Is unit circle a Subspace? Suppose $V$ = $\mathbb{R}^2$ be our vector space over a real field and $W \subset V$, where the subset $W$ = $\{ (x, y) \in \mathbb{R}^2$  $x^2 + y^2 = 1 \}$ is a unit circle in a
Hence, it is not a subspace. 17 
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Figure 5: A unit circle 
BasisA nonempty subset $B$ of a vector space $V$ over the field $F$ is said to be a basis if $B$ is linearly independent over $F$ and spans the vector space $V$. The set of all the vectors that are the linear combination of the vectors in the set V = $\{ v_1, v_2, . . . . . . . . v_n\}$ is called the span of V.
The condition of one vector being a 'linear combination' of the other is termed as Linear Dependence. A set of vectors $\{ v_1, v_2, . . . . . . . . v_n\}$ is said to be linearly dependent if there are scalars $c_1, c_2, . . . . , c_n$, not all zero, such that
Let $v_1, v_2 \in$ V then these vectors are called linearly independent when $v_1 \neq$ $cv_2$, $\forall c \in F$
OR If $v_1$ does not lie in the span of $v_2$, then these vectors are linearly independent.
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Figure 6: Linear Dependence and Independence of the vectors 
Remark: 1) Suppose $B$ is a basis of the vector space $V$ and $B \subseteq T \subseteq V$ 2) Suppose $Q \subsetneq B$ and $B$ is a basis of $V$, then $\exists$ $w \in B$ such that $w \notin Q$. 3) By the above two points, we can undoubtedly conclude that
Existence of BasisEvery finitedimensional vector space admits a basis. This is because every spanning set contains a Basis. OBSERVE: The new set $B$ spans vector space $V$ and this is because our original set also spanned $V$ and we removed just a few vectors that were already in the span of the preceding vectors.
The number of elements in a basis of a vector space $V$ over a field $F$ is called Dimension of the basis and is denoted as $'n'$.
Example Problem: Let $M = \begin{bmatrix} u & u + v \\ u + v & v \end{bmatrix}$ be a vector space over $\mathbb{R}$. Show that the set $B = \left\{ \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix}\right\}$ is a basis for $M$. Solution: Suppose there are real numbers $u$ and $v$ such that 21 
Applications

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Pause and Ponder1) Is a combination of a line and a plane in $\mathbb{R}^3$ forms a subspace? 2) What will happen if there is a union of two subspaces? Will it be also a subspace? 3) What will you call an empty set of vectors {$\phi$}, dependent or independent? 4) Think about whether the concept of linear independence or dependence can be applied for individual vectors or only for the set of vectors. 24 
References[1] Gallian, J. (2012). Contemporary abstract algebra. Nelson Education. [2] Stover, Christopher and Weisstein, Eric W. "Euclidean Space." From MathWorldA Wolfram Web Resource. [3] Wallace, T., Sekmen, A., & Wang, X. (2015). Application of subspace clustering in dna sequence analysis. Journal of Computational Biology, 22(10), 940952. [4] Margalit, D., & Rabinoff, J. (2018). Interactive Linear Algebra. Georgia Institute of Technology. [5] https://link.springer.com/content/pdf/10.1007%2F9783319110806_2.pdf [6] http://homepage.math.uiowa.edu/~roseman/m33/m33_chap_2_sec_8.pdf [7] http://math.mit.edu/~trasched/18.700.f11/lect5article.pdf [8] https://www.math.csi.cuny.edu/~ikofman/review_exam2_F17.pdf 25 
Further Reading[1] http://mathonline.wikidot.com/theintersectionandunionofsubspaces [2] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4589114/ [3] http://math.mit.edu/~trasched/18.700.f11/lect5article.pdf 26 
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